Real-time imaging system for monitoring and control of thermal therapy treatments

ABSTRACT

Methods and systems of monitoring and controlling thermal therapy treatments. One method includes determining, with an electronic processor, a propagation constant for a plurality of waveports. The method also includes modeling an object within an imaging cavity using a sparse mesh model. The method also includes determining a simulated impedance of the object based on the model of the object. The method also includes calibrating the simulated impedance with a theoretical impedance numerically calculated for the object. The method also includes determining a distribution of an electric field and a distribution of a magnetic field for a mode in a conformal modeled waveport based on the calibrated impedance and the model of the object. The method also includes exciting the plurality of waveports to generate the determined distribution of the electric field and the distribution of the magnetic field.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application is a divisional of U.S. patent application Ser. No.15/658,305, filed on Jul. 24, 2017, which is a non-provisional of andclaims priority to U.S. Provisional Patent Application No. 62/365,927,filed on Jul. 22, 2016, the entire contents of each of which are herebyincorporated by reference.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application is a non-provisional of and claims priority to U.S.Provisional Patent Application No. 62/365,927, filed on Jul. 22, 2016,the entire contents of which is hereby incorporated by reference.

BACKGROUND OF THE INVENTION

Thermal therapy is a fast-growing and increasingly versatile category oftreatments in the fight against cancer. Researchers have developed anumber of methods (both invasive and noninvasive) to deliver thermalenergy in a variety of forms (including ultrasound, radiofrequency,microwave, and laser). Common to all of these thermal therapies,however, is the necessity for real-time thermal monitoring and guidance.The current standard of MRI-based thermal monitoring is cumbersome,expensive, and requires the therapy system design to be MR-compatible.

SUMMARY OF THE INVENTION

A real-time noninvasive thermal monitoring system based on multi-staticmicrowave imaging is a potential alternative to MR-based thermalmonitoring. This system uses microwaves to detect the small changes inthe complex permittivity of water that occur due to changes intemperature, and build upon methods and hardware developed previouslyfor the application of breast cancer screening.

Embodiments described herein relate to methods and systems for real-timeimaging of differential temperature in a sample using microwaves, and,in particular, to real-time microwave imaging systems for monitoring andcontrol of thermal therapy treatments.

Commercial applications for some embodiments described herein include,for example, thermal dose monitoring and treatment guidance for tumorablation systems (for example, cryoablation, High Intensity FocusedUltrasound (HIFU), laser induced thermal therapy, radio frequency (RF)ablation, and microwave ablation) and hyperthermia systems. In addition,potential commercial applications exist in the emerging fields of imageguided, thermal drug delivery, and thermally controlled regulation oftransgene expression using heat-sensitive promoters.

The embodiments described herein address a number of issues with currentstate of the art MRI-based monitoring systems, which are limited byexpense, access, size, and the requirement that the therapy system beMRI compatible.

RF and HIFU thermal ablations are performed without spatial temperaturecontrol, currently under computed tomography (CT) or ultrasound (US)monitoring. With US monitoring, the estimation of the actual size of theablated region is not precise because the extent of the hyperechogeniczone does not strictly correspond to the necrotic area. Using CTmonitoring, there is no density change related to acute necrosis.Therefore, for both US and CT monitoring, accurate prediction of thetreatment zone requires intravenous injection of contrast material (forexample, microbubbles or an iodine compound, respectively). Due to theimpracticality of repeated injections, the extent of the ablative zonemay only be assessed at the end of the procedure.

Thus far, real-time thermal monitoring has been limited to thermalprobes or MR thermometry. Thermal probes are fast and accurate. However,the coverage of thermal probes is limited to the sensor tip and isinherently invasive. MRI has been employed for thermal monitoring,especially in ultrasound ablation and microwave hyperthermia. MRIprovides full coverage of a treatment region. However, MRI is limited byexpense, access, and the requirement that the thermal delivery system beMRI compatible.

Ultrasonic thermal monitoring has received attention in shear-wave,speed of sound, backscatter variation, and optoacoustic thermal imaging.However, ultrasonic thermal monitoring has not been clinically proven.

A closed-loop, noninvasive focused microwave thermal therapy system hasbeen developed. The system is capable of safe, rapid, and preciseadjuvant hyperthermia, neoadjuvant hyperthermia, and thermal ablation.Innovations in thermal monitoring and focusing enable improved outcomesin the fastest growing segment of the nonvascular interventionalradiology device market.

Thermal therapy has primarily been used to increase the effectiveness ofother treatment methods (for example, hyperthermia to enhancechemotherapy). However, recently thermal therapy has also been used todeliver localized therapeutic agents to a target region as well as forthermal ablation. Hyperthermia employs a thermal dose that is generallysafe for normal cells (for example, ˜43° C.). However, hyperthermia hasthe potential for the following therapeutic effects: (1) thermal damageor weakening of cancer cells; (2) increased blood flow through weakenedtumor (enhanced delivery of therapeutic agents); (3) increased oxygenlevels in tumors; (4) stimulated immune response; and (5) thermalrelease of therapeutic agents encapsulated in temperature sensitiveliposomes. Thermal ablation employs more extreme temperature levels (forexample, >60° C. for RF, microwave and <0° C. for cryoablation) todestroy tissue. Localization of the damage is achieved using an invasiveprobe (radiofrequency ablation) or catheter (cryoablation), or throughfocusing (HIFU, microwave).

Existing thermal therapy systems are invasive, have poor focusing andcontrol, and require mixed modalities for temperature monitoring.Currently, no wave-based thermal therapy system, or combination ofsystems, can yet deliver highly controlled non-invasive heatlocalization with real-time non-invasive temperature monitoring in thesame system. These limitations have so far prevented wide scale adoptionof microwave thermal therapy despite growing interest in microwavethermal therapy.

At this time, the main application for thermal therapy systems is toimprove local control while limiting radiation doses using concurrenthyperthermia with re-irradiation therapy. Hyperthermia has beendemonstrated to enhance radiation-induced cell killing. Additionally,several randomized trials have shown an improvement in local controlwhen combined with re-irradiation therapy. By potentiating the effect ofradiation, hyperthermia may potentially result in similar tumor controlrates with lower overall doses, thereby decreasing the risks associatedwith re-irradiation therapy. There has also been interest in usingneoadjuvant hyperthermia to reduce tumor size prior to surgery. Beyondhyperthermia applications, ablative treatments have been gainingincreasing attention as an alternative to standard surgical therapies,as thermal ablation has the potential to reduce morbidity and mortalityin comparison with standard surgical resection and has the ability totreat patients with contraindication or those who refuse open surgery.Lastly, thermal therapy has the potential for use in locally targeteddrug delivery, in the cosmetic market, and there is emerging interest inthe role of thermal devices in triggering an immune-response to cancer.

Current Microwave and Radio Frequency (RF) Thermal Treatment Technology

The technologies of thermal treatment with radio and microwave radiationfall under four categories: (1) invasive treatment; (2) non-invasivetreatment; (3) invasive temperature monitoring; and (4) non-invasivetemperature monitoring. Radio frequency ablation (RFA) uses large gaugeneedles that double as coaxial probes. High intensity radio frequencywaves propagate down the probe and excite strong electric fields at theprobe tip. The localized fields are absorbed by the immediatelysurrounding tissue and heat to ablation. While conceptually simple, theprocedure is invasive and cannot reliably treat tumor margins or largetissue regions. There is no active temperature monitoring. Rather,imaging is used to guide the needle. Also, tissue charring may limit thetherapeutic range as treatment progresses. Current microwave ablationsystems generally use the same approach, only at a higher frequency andsome with the use of additional probes, but all are invasive.Non-invasive hyperthermia systems based on array focusing exist forclinical use. Multiple antennas are used to focus microwave radiationsubcutaneously. These systems, however, use very few antennas, and thushave large or undefined focal regions with limited steering ability.Lack of control and modeling may lead to side effects of skin burningand unwanted hot spots.

To monitor heat deposition, invasive 15-gauge catheter fiber opticsensors have been used. While temperature measurement at the probelocation is accurate, coverage is limited to the sensor tip. Invasivetemperature probes are generally not considered a reasonable long-termsolution. Finally, a non-invasive real-time heat monitoring techniquebased on Proton Resonance Frequency Shift (PRFS) Magnetic ResonanceThermal Imaging (MRTI) has been used in conjunction with microwavefocused therapy. MRI provides full coverage of a heated region, but theexpense and need for two modalities limits the use of the therapy systemto institutions with MRI.

Accordingly, embodiments described herein provide methods and systemsthat integrate electromagnetic modeling with hardware design to achieveboth focused microwave thermal therapy with real-time monitoring in thesame system. Focusing is achieved with numerous independently sourcedantennas (for example, 36 or 48 antennas) that surround the treatmentarea and radiate into an enclosure of known construction. The enclosureallows precise electric field modeling of each antenna. The models areused to determine the exact antenna sourcing in order to position,shape, and steer the focal spot as desired. By this method, 3 cm focalspot sizes with better than 1 cm position accuracy may be guaranteedwhile eliminating unwanted hot spots. The system may focus microwaveradiation at one point or many points in a single session. The finecontrol of the spot location also makes faster scanning possible inorder to achieve uniform hyperthermia over large tissue regions, whichis not achievable by current systems. The use of many antennas increasesthe power output, reducing treatment time.

The same antennas and numerical models are also used to implementreal-time microwave imaging for temperature monitoring. Duringtreatment, signal paths are periodically re-routed in order to recordscattered fields taken off of the tissue being heated. Images are formedof the changes in dielectric properties due to heat. The integration ofthe therapy and monitoring are possible because (1) many more sourcessurround the treatment area than in existing systems and (2) theembodiments described herein use the numeric models for microwavefocusing and image formation in the same hardware paradigm. This enablesimplementation of a two-tier monitoring system for improved safety andthermal control. The first is internal monitoring of the microwavecircuit outputs. The outputs are sampled at millisecond refresh ratesand compared to the model-predicted values for the prescribed focusing.Second, real-time imaging of subcutaneous heat deposition at a one frameper second refresh rate. When either method indicates deviations fromdesired behavior, the system is shut down. As part of this closed-loopfeedback, the images are used to adjust the focal spots in real-time.This is the first known implementation of a non-invasive,single-modality treatment/monitoring system with closed-loop real-timefeedback and control.

Embodiments described herein relate to a focused microwave thermaltherapy system that utilizes a novel method of integrated real-timemicrowave thermal monitoring to precisely adjust the amplitude and phaseoutput of custom high power microwave modules. This integrated solutionenables on-the-fly optimization of focal spot localization, uniformityof heating, and hotspot minimization based on continuous feedback from(1) microwave thermal mapping and (2) output power monitoring.

In recent years, thermal therapies have seen increased use in cases ofsub cutaneous cancers of soft tissue. In general, thermal treatments maybe classified into two types: (1) thermal ablation, which employsextreme temperatures to induce rapid and localized tissue destructionand (2) hyperthermia, which employs moderately elevated temperatures(typically between 40-45° C.) to sensitize tumor cells to the effects ofradiation or chemotherapy. The embodiments described herein may be usedfor either treatment method and offers improvements upon existingsystems in each.

Therapy

Current ablative methods include, for example, cryoablation, highintensity focused ultrasound, radio frequency ablation, and microwaveablation.

Hyperthermia is currently induced using radio frequencies, non-focusedultrasound, and focused microwaves, which have been used as an adjuvantto radiation therapy or chemotherapy.

Monitoring

Existing systems rely on either invasive probes or MR thermometry forthermal monitoring and guidance. The use of fiber optic cathetertemperature sensors dates back decades across a range of treatmentmethods. Invasive probes are fast and accurate. However, the coverage ofinvasive probes is limited to the sensor tip and the invasive probes areinherently invasive. MRI has been extensively studied for thermalmonitoring, especially in ultrasound ablation and microwavehyperthermia. MRI provides full coverage of a treatment region. However,MRI is limited by expense, access, and the requirement that a thermaldelivery system be MRI compatible. Ultrasonic thermal monitoring hasreceived attention in shear-wave, speed of sound, backscatter variation,and optoacoustic thermal imaging.

Features of the embodiments described herein include: (1) integratedthermal monitoring, which eliminates the need for additional hardware tomonitor thermal therapy delivered by focused microwave heating; (2)custom microwave hardware with full per channel control of magnitude andphase, which allows for real-time monitoring and control of focusedmicrowave therapy array; and (3) the coupling of (i) integratedreal-time thermal monitoring with (ii) real-time monitoring and controlof the microwave therapy array, which enables real-time, automatedenhancement of focused microwave therapy for optimum delivery of thermaldose with minimal operator intervention, all in a noninvasive manner.

Advantages of the embodiments described herein include the fact that:(1) existing methodologies require either invasive probes or anadditional imaging system present during the procedure, both of whichcomplicate the procedure, may impede treatment, and require operatorintervention to adjust the treatment accordingly; (2) existingmethodologies either use invasive probes to deliver treatment (RFA) orsuffer from long treatment times (HIFU); and (3) none of the existingmethodologies integrate real time thermal monitoring into adaptivetreatment planning in the quantitative manner that the embodimentsdescribed herein offer. A main consequence is that existing systems aredependent upon operator intervention to adjust treatments in aqualitative way, whereas embodiments described herein use algorithmsbased on the quantitative data (for example, thermal mapping andper-channel microwave power and phase) to automatically adapt inreal-time to achieve the optimal treatment dose.

Commercial applications that may be suitable for the embodimentsdescribed herein include (1) thermal ablation, which employs extremetemperatures to induce rapid and localized tissue destruction; (2)hyperthermia, which employs moderately elevated temperatures (forexample, typically between 40-45° C.) to sensitize tumor cells to theeffects of radiation or chemotherapy; and (3) thermally targeted drugrelease.

Accordingly, one embodiment provides a system for real-time microwaveimaging of differential temperature. The system includes a plurality ofantennas arranged in a three-dimensional array adjacent a cavity intowhich is placed a sample to be analyzed, the plurality of antennasincluding at least one transmit antenna and at least one receiveantenna. The system also includes a vector network analyzer, whereineach of the plurality of antennas is operatively connected to the vectornetwork analyzer. The system also includes a controller operativelyconnected to the plurality of antennas and the vector network analyzer.The controller is configured to collect a plurality of pairwisemeasurements from the plurality of antennas to determine a plurality ofS-parameters, wherein each pairwise measurement includes a transmitantenna and a receive antenna. The controller is also configured tocombine the plurality of S-parameters with a precomputed inversescattering solution to form an image.

Another embodiment provides a method of monitoring in real-time amicrowave thermal therapy. The method includes imaging a treatmentregion to obtain dielectric properties, mapping the dielectricproperties to the treatment region, and conducting the microwave thermaltherapy on the treatment region. The method further includes creating athermal image of the treatment region in real-time and controlling themicrowave thermal therapy on the treatment region in response to thecreated thermal image. Creating the thermal image of the treatmentregion relies upon the temperature dependency of the dielectricproperties in the treatment region. Prior to conducting the microwavethermal therapy, the treatment region is positioning within a low-lossdielectric material. The microwave thermal therapy does not have to beMRI compatible for creating a thermal image of the treatment region inreal-time. Creating the thermal image of the treatment region inreal-time also does not require intravenous injection of contrastmaterial and does not require an invasive thermal probe. The microwavethermal imaging data is acquired with a through channel connection tosignal a discrete measurement within a continuous data stream (forexample, hardware) and real-time magnitude thresholding to parse thecontinuous data stream into discrete measurement groups and real-timemedian filtering within each discrete measurement group (for example,software). A portion of the treatment region of which the thermal imageis created may be user-selectable.

Another embodiment provides a system of monitoring in real-time athermal therapy. The system includes an imaging cavity including aplurality of antennas arranged in a three-dimensional array around theimaging cavity. The system also includes a controller operativelyconnected to the imaging cavity. The system is configured to collect,with an electronic processor, a plurality of pairwise measurements fromthe plurality of antennas to determine a plurality of S-parameters. Thesystem is also configured to apply in real-time a combination ofdirect-path triggering signals and magnitude thresholding, along withreal-time median filtering and per-channel magnitude and phasecalibration within each discrete measurement group, to determine aseries of discrete S-parameter measurements from a continuous datastream. The controller is also configured to determine an incidentelectric field inside the imaging cavity for each of the plurality ofantennas. The controller is also configured to generate a numericalfunction directly linking an unknown material property of an object ofinterest to a measured voltage when the object of interest is positionedwithin the imaging cavity. The controller is also configured to update amaterial property of the object of interest. The controller is alsoconfigured to determine a total electric field inside the imaging cavitybased on the updated material property. The controller is alsoconfigured to output an estimated map of complex permittivity for theobject of interest based on the total electric field.

Another embodiment provides a method of monitoring in real-time athermal therapy. The method includes imaging a treatment region toobtain dielectric properties. The method also includes mapping thedielectric properties to the treatment region. The method also includesconducting the thermal therapy on the treatment region. The method alsoincludes creating a thermal image of the treatment region in real-timeand controlling the thermal therapy on the treatment region in responseto the thermal image.

Another embodiment provides a method of monitoring in real-time athermal therapy. The method includes determining, with an electronicprocessor, a propagation constant for a plurality of waveports. Themethod also includes modeling an object within an imaging cavity using asparse mesh model. The method also includes determining a simulatedimpedance of the object based on the model of the object. The methodalso includes calibrating the simulated impedance with a theoreticalimpedance numerically calculated for the object. The method alsoincludes determining a distribution of an electric field and adistribution of a magnetic field for a mode in a conformal modeledwaveport based on the calibrated impedance and the model of the object.The method also includes exciting the plurality of waveports to generatethe determined distribution of the electric field and the distributionof the magnetic field.

Other aspects of the invention will become apparent by consideration ofthe detailed description and accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a cavity-imaging system for differential temperatureimaging with switching matrix, Vector Network Analyzer, Labview control,and fluid circulation.

FIG. 2 is a graph of relative permittivity and conductivity of distilledwater as a function of temperature at 915 MHz, measured with the Agilent85070E dielectric probe.

FIG. 3A illustrates a calibration diagram and instrumentation setup thatincludes a 36-port unknown thru.

FIG. 3B illustrates a fluid filled cavity used as the known thrustandard.

FIG. 3C illustrates a 3×1:12 power divider network tested as an unknownthru.

FIG. 4 illustrates a HFSS CAD model of the cavity, the antenna shadingdesignates transmitter and receiver, and the fluid is filled to 0.5 cmbelow the top.

FIG. 5 illustrates cross cuts of log(|

_(e)(E_(z))|) of the normalized incident field for an antenna in themiddle row, the walls are PEC, and the top surface radiates to the air.

FIG. 6A is a graph of measured and HFSS incident S-parameters includingmagnitude, where symmetric combinations are overlaid and grouped byrow-row interaction.

FIG. 6B is a graph of measured and HFSS incident S-parameters includingphase, where symmetric combinations are overlaid and grouped by row-rowinteraction.

FIG. 7A is a graph of VNA data stream of a single 2-port S-parametermeasurement while the switching matrix cycles over all 324 T/Rcombination, sweep is 4000 points, 2.1 seconds, at 915 MHz including S₁₁(transmitters) and S₂₂ (receivers).

FIG. 7B is a graph of VNA data stream of a single 2-port S-parametermeasurement while the switching matrix cycles over all 324 T/Rcombination, sweep is 4000 points, 2.1 seconds, at 915 MHz including S₂₁and S₁₂ (30 dB differential from RCVR bypass).

FIG. 7C illustrates an inset of FIG. 7A.

FIG. 7D illustrates an inset of FIG. 7B.

FIG. 8A illustrates an example of segmentation including real andimaginary parts of S₂₁, large circles are transition samples.

FIG. 8B illustrates an example of segmentation including threshold ofabsolute value of next-neighbor discrete difference.

FIG. 9A illustrates a control target: water-filled ping-pong ball withembedded thermistor, shown here cooling in the cavity from approximately55° C. to 22.5° C.

FIG. 9B illustrates a plot of ping-pong ball control target temperaturevs. time in the cavity, the dotted line is the ambient temperature ofthe cavity fluid of 22.5° C.

FIG. 10 illustrates an image of |ΔO(r)| after the target has cooled from55° C. to 22.5° C., the initial heated object is taken as the backgroundobject at time zero.

FIG. 11 illustrates the time series of |ΔO(r)| for the water target ofFIG. 10 as it cools from 55° C. to 22.5° C. over 10 minutes, the imagesare formed every 4 seconds, the heated sphere at time zero is taken asthe background object, and cuts are through the peak contrast.

FIG. 12A illustrates cluster of four targets where only the left targetis heated.

FIG. 12B illustrates targets in the cavity.

FIG. 12C illustrates |ΔO(r)| after the water target in the clutter hascooled from 55° C. to 22.5° C., the cluster with heated target is takenas the background object at time zero.

FIG. 13 illustrates the time series of |ΔO(r)| for the water target inclutter of FIG. 12C as it cools from 55° C. to 22.5° C. over 10 minutes,cuts are through the peak contrast.

FIG. 14A illustrates a double chambered breast phantom with vegetableoil and coupling fluid, the heated target placed in inner chamber.

FIG. 14B illustrates the breast phantom of FIG. 14A in the cavity.

FIG. 14C illustrates |ΔO(r)| after the water sphere has cooled, onlydifferential temperature is detected.

FIG. 15 illustrates the time series of |ΔO(r)| for the phantom of FIG.14C, phantom and heated target are taken as the background object attime zero, cuts are through the peak contrast.

FIG. 16 is a graph of normalized pixel intensity at the center of thedetected target in FIGS. 11, 13, and 15 at every time step plottedagainst the normalized temperature of the control target in FIG. 9B,pixel intensity rate of change tracks 1:1 with temperature.

FIG. 17A illustrates a forward model of a 1 cm heated region at a topview of subject's brain and a table of permittivity and conductivityvalues for several different temperatures.

FIG. 17B illustrates a forward model of a 1 cm heated region at a sideview of the subject's brain and a table of permittivity and conductivityvalues for several different temperatures.

FIGS. 18A-B illustrate sample or baseline electric field levels in anunheated condition for a subject's brain (top view).

FIGS. 18C-D illustrate sample or baseline electric field levels in anunheated condition for a subject's brain (front view).

FIGS. 18E-F illustrate sample or baseline electric field levels in anunheated condition for a subject's brain (side view).

FIG. 19A illustrates an image reconstruction of dielectric scatteringand the predicted temperature at a first temperature of 50° C.

FIG. 19B illustrates an image reconstruction of dielectric scatteringand the predicted temperature at a ΔT of 13° C. above body temperature.

FIG. 20A illustrates an image reconstruction of dielectric scatteringand the predicted temperature at a second temperature of 60° C.

FIG. 20B illustrates an image reconstruction of dielectric scatteringand the predicted temperature at a ΔT of 23° C. above body temperature.

FIG. 21A illustrates an image reconstruction of dielectric scatteringand the predicted temperature at a third temperature of 70° C.

FIG. 21B illustrates an image reconstruction of dielectric scatteringand the predicted temperature at a ΔT of 33° C. above body temperature.

FIG. 22A illustrates an image reconstruction of dielectric scatteringand the predicted temperature at a fourth temperature of 80° C.

FIG. 22B illustrates an image reconstruction of dielectric scatteringand the predicted temperature at a ΔT of 43° C. above body temperature.

FIG. 23 graphically illustrates samples taken with a hybrid data streamsegmentation method.

FIG. 24 is a CAD drawing of a prototype imaging cavity.

FIG. 25 illustrates an inversion flow chart with CFDTD forward Solverand Integrated numerical Green's function.

FIG. 26 illustrate effective materials (Row 1: ϵ, Row 2: μ, Row 3: σ)used to model antennas in imaging cavity with CFDTD.

FIG. 27A illustrates |Re(E_(z))| due to single transmitting antenna atresonance at a top view of horizontal cut (xy) through cavity.

FIG. 27B illustrates |Re(E_(z))| due to single transmitting antenna atresonance at a side view of vertical cut (xz) through cavity.

FIGS. 28A and 28B illustrate scattered fields comparison between FDTDsolver and numerical Green's function estimation, magnitudes agreewithin a mean absolute difference of 0.2 dB and phases agree within amean absolute difference of 2 degrees.

FIGS. 29A and 29B illustrate a Yee cell and conformal modeling ofcurvilinear objects where the red square represents the origin of theYee cell.

FIG. 30A illustrates a sparse mesh to model the Teflon filled coaxialfeed.

FIG. 30B illustrates a dense mesh to model the Teflon filled coaxialfeed.

FIGS. 31A-31F illustrate effective material values computed usingEquation (36) for maximally sparse mesh modeling of coaxial feed andindices indicate the number of Yee cell.

FIGS. 32A-32F illustrate conformal meshing of a tapered patch antenna.

FIG. 33 illustrates a waveport for S-parameter extraction.

FIGS. 34A and 34B illustrate wave propagation and modeled impedance ofcoaxial cable.

FIG. 35 illustrates propagation constant of the first five modes of therectangular waveguide.

FIGS. 36A-36D illustrate a cross-sectional field distribution of coaxialfeed TEM mode and dense mesh model results are shown for illustrationpurposes, but sparse mesh results are equally accurate.

FIGS. 37A-37B illustrate tapered patch antenna modeling; magnituderesults agrees with a mean absolute difference of 0.17 dB and a maximumdifference of 0.52 dB and phase results agree with a mean absolutedifference of 2.1 degrees and a maximum difference of 9.8 degrees.

FIGS. 38A and 38B illustrate coaxial probes in a rectangular waveguide.

FIGS. 39A and 39B illustrate the S21 of empty waveguide with coaxialprobes; magnitude results agree with a mean absolute difference of 0.31dB and a maximum difference of 0.72 dB and phase results agrees withmean absolute difference of 5.7 degrees and a maximum difference of 10.4degrees.

FIG. 40 illustrates a waveport source.

FIG. 41 illustrates a CAD model for simulation region of Example 1 wherethe two rectangular air-filled aperture of the size of WR-340 (left) andWR-430 (right) are embedded into the PEC wall.

FIGS. 42A and 42B illustrate a comparison of scattered S21 simulated byFDTD and predicted by numerical vector Green's function; the amplitudeagrees within 0.5 dB and phase agrees within 3 degrees.

FIG. 43A illustrates a dielectric loaded two-probe fed waveguide at atop view of the waveguide and the relative location of the nylon blockwithin the waveguide.

FIG. 43B illustrates a dielectric loaded two-probe fed waveguide at aside view of the waveguide.

FIG. 44 illustrates a CAD model of the dielectric loaded waveguide wherea waveport source is used to excite the coaxial feed.

FIGS. 45A and 45B illustrate a comparison of scattered S21 measured withVNA, simulated by FDTD and predicted by numerical vector Green'sfunction. Compare the simulated and predicted S21, the amplitude agreeswithin 0.5 dB and the phase agrees within 2 degrees. Compare themeasured and predicted S21, the amplitude agrees within 2 dB and thephase agrees within 10 degrees.

FIGS. 46A and 46B illustrate an imaging cavity and dimensions of atapered patch antenna.

FIG. 47A illustrates a CAD of the imaging cavity with the nylon block,where waveport 1, 2 and 3 are located in the middle ring of antenna feedand waveport 4 is located in the bottom ring of antenna feed.

FIG. 47B illustrates a CAD of the imaging cavity with the nylon block ata top view of the cavity with the location of the 4 coaxial feedwaveports, where waveport 1, 2, and 3 are located in the middle ring ofantenna feed and waveport 4 is located in the bottom ring of antennafeed.

FIGS. 48A and 48B illustrate a comparison of scattered S21; theamplitude and phase of the simulated and predicted S21 agrees within 0.2dB and 2 degrees and the amplitude and phase of the measured andpredicted S21 agrees within 2 dB and 8 degrees.

FIGS. 49A and 49B illustrate a comparison of scattered S31; theamplitude and phase of the simulated and predicted S31 agrees within 0.2dB and 2 degrees and the amplitude and phase of the measured andpredicted S31 agrees within 2 dB and 10 degrees.

FIGS. 50A and 50B illustrate a comparison of scattered S41; theamplitude and phase of the simulated and predicted S41 agrees within 0.2dB and 2 degrees and the amplitude and phase of the measured andpredicted S41 agrees within 0.5 dB and 10 degrees.

DETAILED DESCRIPTION

One or more embodiments are described and illustrated in the followingdescription and accompanying drawings. These embodiments are not limitedto the specific details provided herein and may be modified in variousways. Furthermore, other embodiments may exist that are not describedherein. Also, the functionality described herein as being performed byone component may be performed by multiple components in a distributedmanner. Likewise, functionality performed by multiple components may beconsolidated and performed by a single component. Similarly, a componentdescribed as performing particular functionality may also performadditional functionality not described herein. For example, a device orstructure that is “configured” in a certain way is configured in atleast that way, but may also be configured in ways that are not listed.Furthermore, some embodiments described herein may include one or moreelectronic processors configured to perform the described functionalityby executing instructions stored in non-transitory, computer-readablemedium. Similarly, embodiments described herein may be implemented asnon-transitory, computer-readable medium storing instructions executableby one or more electronic processors to perform the describedfunctionality.

In addition, the phraseology and terminology used herein is for thepurpose of description and should not be regarded as limiting. Forexample, the use of “including,” “containing,” “comprising,” “having,”and variations thereof herein is meant to encompass the items listedthereafter and equivalents thereof as well as additional items. Theterms “connected” and “coupled” are used broadly and encompass bothdirect and indirect connecting and coupling. Further, “connected” and“coupled” are not restricted to physical or mechanical connections orcouplings and can include electrical connections or couplings, whetherdirect or indirect. In addition, electronic communications andnotifications may be performed using wired connections, wirelessconnections, or a combination thereof and may be transmitted directly orthrough one or more intermediary devices over various types of networks,communication channels, and connections. Moreover, relational terms suchas first and second, top and bottom, and the like may be used hereinsolely to distinguish one entity or action from another entity or actionwithout necessarily requiring or implying any actual such relationshipor order between such entities or actions.

Section I: Microwave Imaging System for Real-Time Monitoring andFeedback of Thermal Therapy Systems

Embodiments described herein relate to a microwave imaging system forreal-time 3D imaging of differential temperature for the monitoring andfeedback of thermal therapy systems. Design parameters are constrainedby features of a prototype focused microwave thermal therapy system forthe breast, operating at 915 MHz. Real-time imaging is accomplished witha precomputed linear inverse scattering solution combined withcontinuous Vector Network Analyzer (VNA) measurements of a 36-antenna,HFSS modeled, cylindrical cavity. Volumetric images of differentialchange of dielectric constant due to temperature are formed with arefresh rate as fast as 1 frame per second and 1° C. resolution.Procedures for data segmentation and postprocessed S-parametererror-correction are developed. Antenna pair VNA calibration isaccelerated by using the cavity as the unknown thru standard. The deviceis tested on water targets and a simple breast phantom. Differentiallyheated targets are successfully imaged in cluttered environments. Therate of change of scattering contrast magnitude correlates 1:1 withtarget temperature.

In recent years, thermal therapies have seen increased use in thetreatment of a variety of diseases, particularly in cases ofsubcutaneous cancers of soft tissue. In general, these treatments may beclassified into two types: (1) thermal ablation, which employs extremetemperatures to induce rapid and localized tissue destruction and (2)hyperthermia, which employs moderately elevated temperatures (forexample, typically between 40-45° C.) to sensitize tumor cells to theeffects of radiation or chemotherapy. Ablative methods includecryoablation, high intensity focused ultrasound, radio frequencyablation, and microwave ablation. Hyperthermia is typically inducedusing radio frequencies, non-focused ultrasound, and focused microwaveswhich have been used as an adjuvant to radiation therapy orchemotherapy.

As thermal treatment systems evolve, the need for thermal monitoring hasmotivated research in several modalities. The use of fiber opticcatheter temperature sensors dates back decades across a range oftreatment methods. Probes are fast and accurate. However, the coverageof probes is limited to the sensor tip and probes are inherentlyinvasive. MRI has been extensively studied for thermal monitoring,especially in ultrasound ablation and microwave hyperthermia. MRIprovides full coverage of a treatment region, but is limited by expense,access, and the requirement that a thermal delivery system be MRIcompatible. Ultrasonic thermal monitoring has received attention inshear-wave, speed of sound, backscatter variation, and optoacousticthermal imaging. Also, microwave thermal imaging in a medical settinghas been reported for two-dimensional (2D) antenna geometries withinverse scattering techniques. The same system was used to monitor HIFUwith 35 second refresh rates. Finally, monitoring provides a mechanismfor controlled feedback, which has been reported using real-time MRIguided microwave hyperthermia.

Embodiments described herein relate to microwave-based real-time thermalmonitoring within the constraints of focused microwave thermal therapysystem parameters in order to avoid the cost and complexities of dualmodality implementations. This is done within the design parameters of apreviously developed pre-clinical focused microwave prototype for breastcancer therapy. As illustrated in FIG. 1 , the imaging system 100 iscomposed of a cavity-like antenna array at 915 MHz with coupling fluid,vector network analyzer (VNA), and solid-state switching matrix. Using aVNA in this context, with experimental features developed to do so,obviates the need for custom radar hardware in order to accomplishreal-time thermal monitoring. Note that this system, as presented, isdesigned to image changes in dielectric constant due to the applicationof microwave heating, rather than image the structure of the breast fordiagnostic purposes.

Microwave imaging of differential temperature is predicated on thechange in dielectric properties of water with temperature. It is wellknown that the dielectric properties of water change as a function oftemperature. The relative permittivity and conductivity of distilledwater as a function of temperature at 915 MHz are plotted in FIG. 2 .Within the range of thermal therapy temperatures (for example, 35°C.-60° C.), the relative permittivity of water changes by as much as afew units, and the conductivity up to a factor of 2. The dielectricchange of animal liver tissue has been reported to be approximately 1%per ° C. between 28° C. and 53° C. at 915 MHz. Studies of excised breasttissue show that soft-tissue dielectric properties are dominated bywater fraction, and are expected to change proportionally withtemperature. Thus, even though the dielectric change is small, theability to detect the dielectric change has been reported in bothmedical and non-medical settings.

Methods

A. Imaging System

As illustrated in FIG. 1 , the imaging system 100 includes an imagingenclosure 105. The imaging enclosure 105 is a cavity. As illustrated inFIG. 1 , the imagining enclosure 105 includes twelve vertical panels ofmicrowave substrate soldered to each other and to a conducting base.Opposing panels are separated by 15 cm, and the cavity is 17 cm tall.There are three antennas per panel for a total of 36 antennas, designedto operate at 915 MHz in a coupling fluid. 2-port S-parametermeasurements are taken with an Agilent PNA-5230A Vector Network Analyzer(VNA) between antenna pairs. Pair-wise measurements are routed through18 transmit (T) antennas and 18 receive (R) antennas using two SP18Tsolid state switching matrices that connect to Ports 1 and 2,respectively, of the VNA, providing 182=324 pair-wise measurements. T/Rantennas alternate around the rows of the cavity and between the rows,as shown in FIG. 4 .

The coupling medium is a mixture of isopropyl alcohol and water designedto couple microwave energy into breast tissue. Other fluids have beeninvestigated in literature, such as safflower oil, oil/water emulsion,glycerin, and acetone. The medium is usually chosen to balanceconductive loss, relative permittivity, toxicity, antenna match, andease of use. An alcohol/water mixture is inexpensive, clean,controllable, and not prohibitively lossy at 915 MHz. A 20:7alcohol:water ratio is used. The dielectric properties of the fluid aremeasured with the Agilent 85070E dielectric probe kit, at 915 MHz therelative permittivity and conductivity are (ε_(γ), σ)=(24.5, 0.460 S/m).

Because the medium is a mixture, the medium must be circulated toprevent separation. An external aquarium pump recirculates the fluidthrough a drain in the bottom of the cavity to a nozzle at the top. Thedrain consists of seven 3 mm holes, mimicking a wire mesh, and thus acontinuous boundary at 915 MHz. The fluid exchanges once per minute.

The antennas are bow-tie patch antennas designed originally to operateat 915 MHz in an oil/water emulsion with (ε_(γ), σ)=(23, 0.05 S/m). Inthe current coupling medium, the resonance remains, but the impedancematch is degraded to about −6 dB.

The SP18T switches each consist of one SP16T switch with twoMinicircuits ZYSWA-2-50DR SPDT switches connected to two of the paths.The ensemble is controlled via a computer parallel port. Path isolationis better than 60 dB, except for the Minicircuits switch paths, whichare isolated to 40 dB. Differences in attenuation between individualpaths are not critical, because each path is individually calibrated.

To maximize the signal to noise ratio (SNR), +13 dBm output power isused, which is the maximum leveled output power of the VNA. In addition,the VNA's 2-port test set is configured to bypass the Port 2 receiver(RCVR) coupler. This increases the Port 2 receiver sensitivity, and thusS₂₁ measurements, by +15 dB, while degrading S₁₂ by −15 dB. However,only S₂₁ is used for imaging.

B. Imaging Cavity Used as Unknown ‘Thru’

In order to compare cavity S-parameter measurements to HFSS and forwardscattering predictions (below), a unique 2-port calibration is requiredbetween every T/R combination, 324 in total. To accelerate the process,the 2-port unknown ‘thru’ calibration is used, wherein the cavity itselfprovides ‘thru’ paths for every T/R pair. Thru measurements are splicedwith individual 1-port calibrations to complete the calibration sets(calsets), diagrammed in FIGS. 3A-3C.

The calibration procedure follows:

-   -   1) 1-port Short-Open-Load (SOL) calibrations are saved for each        of the 18 transmit and 18 receive paths.    -   2) The paths are connected to the fluid-filled cavity.    -   3) A Labview routine selects and switches to a T/R combination.    -   4) The VNA loads the 1-port SOL calibrations for that        combination, completes the unknown thru measurement, computes        the error terms, and saves the calset.    -   5) Process repeats at 3).

Excluding time taken for 1-port SOL calibrations, all 324 2-port calsetsare completed in 5 minutes.

Unknown thru calibration is accurate to within the sign of S₁₂ and S₂₁.The sign ambiguity is resolved by having rough knowledge of thecorrected thru measurement phase by, for example, comparing the phase ofcorrected S₁₂ and S₂₁ to that predicted by the HFSS cavity model.

The method is validated by comparing cavity measurements collected withtwo calibration sets. The first set uses the fluid-filled cavity as thethru, the second set uses a 36-port power-divider network, as shown inFIGS. 3B and 3C, respectively. The divider network is created byconnecting three Minicircuits ZN12PD-252-S+1:12 power dividers with atee. The power divider thru provides nearly equal transmission betweenall ports, which is not the case for the cavity. Data collection,switching, and calset loading are all automated with Labview. Cavitymeasurements of the incident field S-parameters using either calibrationset are identical to within a sign of S₁₂ and S₂₁, confirming thateither the cavity or power divider may be used as a multi-way unknownthru standard.

In general, any 36-port device with reasonable transmissioncharacteristics may be used as the thru. The advantage of using thecavity itself is that the thru measurements may b e repeated withoutdisassembling the setup. However, recalibration is only necessary whenthe system configuration changes. Overall, this addresses for the firsttime the unsolved problem in microwave medical imaging of how toefficiently and repeatedly calibrate multi-sensor systems for VNAmeasurements.

C. HFSS Numerical Model

Ansys HFSS may be used to numerically model the cavity, which requiresmatching the S-parameter reference planes between simulation andexperiment. HFSS may be used to estimate the incident fields throughoutthe cavity, which include all background multiple scattering not presentbetween the object and the cavity, and are used in the forwardscattering model described below. FIG. 4 shows the HFSS CAD model;antennas are shaded as transmitters and receivers. FIG. 5 shows theincident field of one antenna. The drain is assumed to be a continuouscopper sheet.

Measured and simulated incident S-parameters may be used to assess theaccuracy of the HFSS model. FIGS. 6A and 6B show the magnitude andphase, respectively, of the measured and simulated incident S21S-parameters. The plots overlay circular-symmetric T/R paths, groupedaccording to the 6 row-row interaction combinations. The groupings fromleft to right are: middle-middle, middle-top, middle-bottom, top-top,top-bottom, bottom-bottom. The data is plotted this way to assess thesymmetry of both the cavity measurements and HFSS simulation, as well asto compare the cavity measurements and HFSS simulation. In general, themagnitude agrees to within 2 dB, and the phase agrees as well as 5degrees but no worse than 30 degrees. Agreement is best between antennasin the same row.

D. Forward Scattering Model

The forward scattering model predicting the scattered field S-parameterbetween a transmitter i and receiver j in the presence of an object isgiven by:S _(ji,sca) =C∫E _(inc,j)(r′)·O(r′)E _(i)(r′)dV′  Equation (1)where E_(i) is the total object field produced by the transmitter andE_(inc,j) is the incident field of the receiver.

The object function is defined as:

$\begin{matrix}{{0(r)} = {{{\delta\epsilon}(r)} + {i\frac{\delta{\sigma(r)}}{{\epsilon}_{b}\omega}}}} & {{Equation}(2)}\end{matrix}$where ω is the operating frequency in radians and the backgroundpermittivity is ϵ_(b)=ϵ_(o)ϵ_(yb) with relative permittivity ϵ_(yb).

The contrasts are given by:

$\begin{matrix}{{\delta{\epsilon(r)}} = {\frac{\epsilon(r)}{\epsilon_{b}} - 1}} & {{Equation}(3)}\end{matrix}$ $\begin{matrix}{{\delta{\sigma(r)}} = {{\sigma(r)} - \sigma_{b}}} & {{Equation}(4)}\end{matrix}$where σ_(b) is the background conductivity, δϵ(r) is unit-less and δσ(r)is an absolute measure of conductivity with units of S/m.

The constant is given by:

$\begin{matrix}{C = {- \frac{k_{0}^{2}Z_{0}^{j}}{2i\omega\mu a_{0}^{i}a_{0}^{j}}}} & {{Equation}(5)}\end{matrix}$where k₀ is the lossless background wavenumber, k₀ ²=ω²μ₀ϵ_(b), Z_(o)^(j) is the characteristic impedance of the receiver transmission line,and a₀ ^(i) and a₀ ^(j) are the excitations at the S-parameter referenceplanes of the transmit and receive antennas, respectively.

When estimating the cavity fields with HFSS, the constant simplifies. Insimulation, the average transmit power is set to 1 Watt, thus the linevoltage isa ₀=√{square root over (2P _(ave) Z ₀)}=√{square root over (2Z₀)}  Equation (6)where the phase of a₀ is zero because the S-parameter reference planesof the HFSS model and the cavity are the same. Equation (5) reduces to

$\begin{matrix}{C = {- \frac{k_{0}^{2}}{4i\omega\mu}}} & {{Equation}(7)}\end{matrix}$

Finally, in measurement, the scattered field S-parameters are obtainedby subtracting the S-parameters for the total and incident fieldsS _(ji,sca) =S _(ji,tot) −S _(ji,inc)  Equation (8)

To increase the accuracy of Equation (1) when comparing Equation (1)with measurement, both sides may be normalized by the incident fieldS-parameters, while carefully distinguishing simulation frommeasurement:

$\begin{matrix}{\frac{S_{{ji},{sca}}^{meas}}{S_{{ji},{inc}}^{meas}} \cong {\frac{C}{S_{{ji},{inc}}^{HFSS}}{\int{{{E_{{ing},j}^{HFSS}\left( r^{\prime} \right)} \cdot {O\left( r^{\prime} \right)}}{E_{i}^{HFSS}\left( r^{\prime} \right)}{dV}^{\prime}}}}} & {{Equation}(9)}\end{matrix}$This helps eliminate multiplicative systematic errors, such as a phaseshifts.

E. Differential Scattering

Because the change in the dielectric constant of water due totemperature is small, the inverse scattering problem may be formulatedas a perturbation about a background object, where a distorted-typevolume integral equation (VIE) is most natural. Formulating the inversescattering problem this way permits the use of differential total fieldmeasurements between heated and unheated objects, which helps eliminateadditive systematic errors. The distorted VIE may be augmented with thenormalizations and multiplying constant of Equation (9). Also, real-timeimaging relies on precomputing the inverse scattering solution, asdescribed in the next section. This may only been done with a constantfield estimate: two such estimates are provided in this section.

Let the unheated background object be designated O_(b)(r), with totalfield E^(b), and scattered fields S_(ji,sca) ^(b), and let the heatedobjected be O_(h)(r), with total field E^(h), and scattered fieldsS_(ji,sca) ^(h). For the case of a differential object measured againsta background inhomogeneity, Equation (1) becomes:ΔS _(ji,sca) =C∫E _(j) ^(b)(r′)·ΔO(r′)E _(i) ^(h)(r′)dV′  Equation (10)where the volumetric differential scattering contrast between the heatedand unheated objects is given by:ΔO(r)=O _(h)(r)−O _(b)(r)  Equation (11)and the differential scattered field is given by:ΔS _(ji,sca) =S _(ji,sca) ^(h) −S _(ji,sca) ^(b) =S _(ji,tot) ^(h) −S_(ji,tot) ^(b)  Equation (12)

The right side of Equation (12) results from cancellation of theincident S-parameters, where S_(ji,tot) ^(b) is the baseline objectmeasurement and S_(ji,tot) ^(h) changes with temperature. Note thatS_(ji,tot) ^(b) and S_(ji,tot) ^(h) are direct measurements.

Two approximations of Equation (10) are possible. The first is thetraditional Born Approximation (BA), in which the total fields are equalto the incident field. The second is the Distorted Born Approximation(DBA), where E^(h)=E^(b). Both are conceptually similar. In the BA, theincident fields are those of the empty cavity. In the DBA, the incidentfields are those in the background object. Differential scattered fieldsfor the two cases are given by:ΔS _(ji,sca) ^(BA) ≈C∫E _(inc,j)(r′)·ΔO(r′)E _(inc,i)(r′)dV′  Equation(13)ΔS _(ji,sca) ^(DBA) ≈C∫E _(j) ^(b)(r′)·ΔO(r′)E _(i)^(b)(r′)dV′  Equation (14)

The choice of approximation depends largely on the informationavailable. The BA requires simulation of only the empty cavity. The DBArequires knowledge of the total fields in the background object, and,thus, the dielectric properties of the unheated object. These propertiesmay be estimated through a full inverse scattering treatment (forexample, BIM, DBIM, and the like), that simultaneously estimates thetotal fields. Alternatively, the object properties may be inferred (forexample, from an MRI study) after which the total fields are computedwith one pass of a full-wave solver that captures object cavityscattering interactions. In general, the DBA is preferred because DBA isthe appropriate approximation for differential scattered fieldmeasurements. For simplicity, the BA is used in the experimentsdescribed below, which show that the BA yields reasonable results.

F. Precomputed Linear Inverse Scattering Solution

The crux of real-time quantitative imaging is precomputing the inversescattering solution. This may be done with a fixed field approximation,such as before. Linearization consists of discretizing the integralequation, after which the object is estimated by comparing forward modelpredictions to differential scattered field measurements. The costfunction used to measure the difference between forward modelpredictions and data is:F(M)=∥Gm−d∥ _(D) ² +∥m∥ _(M) ²  Equation (15)The vector norms ∥·∥_(D) ² and ∥·∥_(M) ² are defined by the inverse dataand model covariance operators, C_(D) ⁻¹ and C_(M) ⁻¹, respectively. Gis the discretized linear forward operator, the vector d containsS-parameter measurements, and m is a complex vector of pixels of thedifferential object.

The least squares solution of Equation (15) in matrix form is:{tilde over (m)}=(G*C _(D) ⁻¹ G+C _(M) ⁻¹)⁻¹ G*C _(D) ⁻¹ d  Equation(16)where {tilde over (m)} is the object estimate. When the data and imagepixels are separately independent, and the noise and pixel uncertaintiesare constant, the inverse covariance matrices are scalars equal to1/σ_(d) ² and 1/σ_(m) ², respectively. Equation (16) reduces to:{tilde over (m)}=(G*G+λ ² I)⁻¹ G*d  Equation (17)which is the familiar regularized normal equations solution withTikhonov parameter λ=σ_(d)/σ_(m). Given an SVD decomposition G=USV*,Equation (17) may be written:{tilde over (m)}=Md  Equation (18)where:M=V(S ²+λ² I)⁻¹ SU*  Equation (19)

The matrix M, once computed and stored, is applied in real-time to d toobtain a new object estimate.

G. Real-Time Data Processing

Labview and Matlab may be used in tandem to collect the scattered fieldmeasurements and process and display images in real-time. Labviewcontrols the VNA and the switches and Matlab processes the data andcomputes the image. The VNA is triggered to save a single data stream asthe switches cycle through every T/R pair. Matlab actively checks fornew data files and then segments the data stream and produces the image.

-   -   1) VNA data stream: All pair-wise scattering parameters are        collected by the VNA in a single measurement. The VNA triggers a        4000 point, 2.1 second CW measurement at 915 MHz with an IF        bandwidth of 2 kHz. The switches are cycled through all 324 T/R        combinations throughout the duration of the measurement to        produce a data stream of S-parameters for all T/R combinations.        6 milliseconds per path yields approximately 10 points per T/R        pair. In actuality, the VNA triggers twice: the first trigger        sources Port 1 to record S₁₁ and S₂₁, the second trigger sources        Port 2 to record S₂₂ and S₁₂. As a result, the switches are        cycled twice, and an entire data set is collected every 4        seconds. Because the data stream is a single measurement and        captures the response of all T/R paths, S-parameter error        correction must be turned off and applied in post-processing.

FIGS. 7A-7D show the magnitude of a raw incident field data stream ofall four S-parameters. S₂₂ and S₁₂ are retrospectively aligned withS_(ii) and S₂₁. S_(ii) is measuring the reflection coefficients of thetransmit antennas, while S₂₂ measures the reflection coefficients of thereceive antennas. Receivers are cycled per transmitter, hence the 18×18nested pattern in FIG. 7A. Lower reflection coefficient magnitudes aredue to longer uncorrected paths through the SPDT Minicircuits switches.FIG. 7B shows S₂₁ and S₁₂. S₂₁ is approximately 30 dB higher than S₁₂,due to the power level asymmetry created by the Port 2 RCVR couplerbypass and the fact that error correction is off. FIG. 7C shows theinset of FIG. 7A and FIG. 7D shows the inset of FIG. 7B.

-   -   2) S-parameter error correction: As mentioned, error correction        is turned off during the sweep and applied after segmentation        (below). The 12-term error model is shown below. Given        uncorrected 2-port measurements S_(11m), S_(21m), S_(12m), and        S_(22m), the corrected S-parameters computed with the standard        12-term error model are:

$\begin{matrix}{S_{11} = \frac{{A\left\lbrack {1 + {e_{22}^{\prime}D}} \right\rbrack} - {e_{22}{BC}}}{E}} & {{Equation}(20)}\end{matrix}$ $\begin{matrix}{S_{21} = {\frac{B}{E}\left\lbrack {1 + {\left( {e_{22}^{\prime} - e_{22}} \right)D}} \right\rbrack}} & {{Equation}(21)}\end{matrix}$ $\begin{matrix}{S_{12} = {\frac{C}{E}\left\lbrack {1 + {\left( {e_{11}^{\prime} - e_{11}} \right)A}} \right\rbrack}} & {{Equation}(22)}\end{matrix}$ $\begin{matrix}{S_{22} = \frac{{D\left\lbrack {1 + {e_{11}A}} \right\rbrack} - {e_{11}^{\prime}{BC}}}{E}} & {{Equation}(23)}\end{matrix}$where:

$\begin{matrix}{A = \frac{S_{11m} - e_{00}}{e_{10}e_{01}}} & {{Equation}(24)}\end{matrix}$ $\begin{matrix}{B = \frac{S_{21m} - e_{30}}{e_{10}e_{32}}} & {{Equation}(25)}\end{matrix}$ $\begin{matrix}{C = \frac{S_{12m} - e_{03}}{e_{23}e_{01}}} & {{Equation}(26)}\end{matrix}$ $\begin{matrix}{D = \frac{S_{22m} - e_{33}}{e_{23}e_{32}}} & {{Equation}(27)}\end{matrix}$ $\begin{matrix}{E = {{\left\lbrack {1 + {e_{11}A}} \right\rbrack\left\lbrack {1 + {e_{22}^{\prime}D}} \right\rbrack} - {e_{22}e_{22}^{\prime}{BC}}}} & {{Equation}(28)}\end{matrix}$

For convenience, Table I lists the error terms and their correspondingSCPI designations in Agilent PNA-type network analyzers at the time ofpublication.

TABLE 1 Error VNA SCPI Term Description Name e₀₀ Directivity(1,1)EDIR,1,1 e₃₃ Directivity(2,2) EDIR,2,2 e₁₁ SorceMatch(1,1) ESRM,1,1 e′₂₂SourceMatch(2,2) ESRM,2,2 e′₁₁ LoadMatch(1,2) ELDM,1,2 e₂₂ LoadMatch(2,1 ELDM,2,1 e₁₀e₀₁ ReflectionTracking(1,1) ERFT,1,1 e₂₃e₃₂ReflectionTracking(2,2) ERFT,2,2 e₁₀e₃₂ TransmissionTracking(2,1)ETRT,2,1 e₂₃e₀₁ TransmissionTracking(1,2) ETRT,1,2 e₃₀ CrossTalk(1,2)EXTLK,1,2 e₀₃ CrossTalk(2,1) EXTLK,2,1

The error terms for all T/R 2-port calibrations are downloaded withLabview using Agilent VNA-SCPI commands. The 12-term model was crosschecked against VNA-calibrated data sets, including those from theunknown thru tests, with identical results.

Technically, all four S-parameters are required in the 12-term errormodel, but for imaging only S₂₁ was used. In the equation for correctedS₂₁, the contribution of S₁₂ is third order, so it is safely set to zerosince it is already 30 dB lower. In future work, the possibility oftreating S₁₁ and S₂₂ as constant in differential scattering scenarios toreduce data collection time may be investigated. In initial tests notdescribed herein, the data collection has been refined to less than onesecond.

-   -   3) Data stream segmentation: After a new data set is detected        and loaded by Matlab, the data streams for S₁₁, S₂₁, and S₂₂ are        segmented to extract individual T/R measurements as follows:    -   1) Separate real and imaginary parts.    -   2) Apply a fast 1D Total Variation (TV) filter. This preserves        the block-like pattern of the data stream while smoothing the        steps.    -   3) Compute the next-neighbor discrete difference. Take the        absolute value. Spikes denote segment transitions. The TV filter        is necessary to prevent noise from corrupting this step.    -   4) Threshold the combined discrete differences of the real and        imaginary parts.    -   5) Check for 324 segments; discard the data set if fewer are        detected.    -   6) Average the points within a segment, excluding transitions.    -   7) Apply error correction to the segmented T/R data (e.g., the        12-term model).    -   8) Correct the S₂₁ phase ambiguity with the HFSS model        (correction is predetermined).

FIG. 8A illustrates the segmentation of the real and imaginary parts ofa portion of the S21 data stream. Transition points are highlighted andexcluded. FIG. 8B shows the absolute value of the next-neighbor discretedifferences with the threshold. The threshold was chosen by trial anderror.

Because segmentation is based on a discrete-difference, segmentation isrobust to timing inconsistencies. This is important because the exactnumber of points per segment may vary due to VNA triggering, CPUallocation, or parallel port signaling. However, segmentation is nowsensitive to noise, hence the need for the TV filter. Also, countingerrors may occur when two adjacent data are exactly the same.Practically speaking, this is rare and usually due to bit errors, andmay be handled in future versions of this system.

After segmentation and error correction, data are mapped to the vectordin Equation (18) and a new image is formed by multiplying the data withthe precomputed inverse scattering solution. The entire post-processingand image formation sequence takes a fraction of a second.

A new data stream segmentation method has also been developed thatprovides improved reliability and increased speed. This method employs ahybrid hardware/software solution that allows for simple magnitudethresholding to accurately determine each transition between individualantenna pair measurements within the data stream. The hardware componentinvolves employing an additional switching channel that is a directconnection between the transmit switching system and the receivingswitching system (bypassing the imaging cavity). This direct connectionresults in a much higher magnitude received signal (>20 dB) than any ofthe antenna pair measurements (see FIG. 23 ). By switching to thisdirect hardware path between each successive antenna pair measurement,it is possible to implement a very robust and fast parsing scheme usingstraight magnitude thresholding (for example, whenever the receivedsignal exceeds a certain magnitude threshold, a switch is known tooccur).

Results and Discussion

The target used to create a differential change in dielectric propertiesdue to a change in temperature is a 4 cm diameter ping-pong ball that isfilled with water and sealed. This is meant to emulate the size andshape of a thermal therapy focal spot in this cavity at 915 MHz, whichis approximately 3 cm in diameter. The target is heated to approximately55° C. in a water bath and then placed in the cavity and imaged whilethe target cools.

The target is held with a thin plastic straw and suspended from a pieceof rigid, low-dielectric ROHACELL foam as shown in FIG. 9A. As acontrol, the change in temperature may be measured with an embeddedthermistor as the target cools to the ambient cavity temperature ofabout 22.5° C., the plot is shown FIG. 9B. In reality, thetime-dependent temperature profile of the target is more complicatedthan may be represented by a single thermistor measurement. Improvementsto this method are listed in the conclusion section.

In the examples that follow, the thermistor is removed and it is assumedthat the target cools at the same rate as the control. Also, the heatedobject at time zero is taken as the background object. Thus, adifferential change in dielectric properties as the target goes from ahigher temperature to lower temperature may be tracked. This is thereverse of what occurs in most thermal therapy systems. This may be donefor experimental convenience since the temperature dependence ofdielectric constant of water remains the same whether the water heats upor cools down, which may be detected as an overall change in scatteringcontrast.

In addition, although the inversion estimates complex ΔO(r), images of|ΔO(r)|. This is because |ΔO(r)| is a measure of the overall change inscattering contrast relative to the background. It avoids ambiguity insimultaneous inversion of both relative permittivity and conductivity,without the need to assert stringent a priori relations between them. Inits current form, the inversion obtains the correct sign of the complexcontrast for non-differential images of simple targets. For differentialimaging and the BA, the inversion may split the scattering contributiondifferently between the two properties for different objects. This issuemay be improved upon in future versions. Despite this, the relationbetween |ΔO(r)| and the total scattered field power is conserved throughthe L2 norm of the cost function. Thus, it is theoretically sufficientto correlate |ΔO(r)| with temperature to accomplish differential thermalmonitoring provided that the contrast changes. This is verifiedempirically in the following examples.

In all the examples that follow, 3D images of |ΔO(r)| are computed every4 seconds, pixelated at 2 mm. The imaging regions are cylindrical, andthe images are not interpolated.

Example 1: One 4 cm isolated water-filled ping-pong ball is imaged asthe ball cools from 55° C. to about 22.5° C. This is the same ball asshown in FIG. 9A without the thermistor. The heated ball is taken as thebackground object and the BA is appropriate for the field estimate. A 3Dslice view of the final image in the sequence, after the ball hasreached the ambient cavity temperature, is shown in FIG. 10 . Thisrepresents the maximum differential change in the scattering contrast. Atime series at {1,4,7,10} minutes is shown in FIG. 11 . The magnitude ofthe contrast clearly increases and settles after 10 minutes. The sphereis well resolved, demonstrating a resolution of 3-4 centimeters, whichis on the order of the thermal therapy focal spot size in this cavity.

Example 2: The same 4 cm water-filled ping-pong ball is heated andimaged as it cools while in the presence of three other objects. Theseobjects consist of two water filled ping-pong balls and a 1 inchdiameter acrylic sphere that act as clutter shown in FIG. 12A. Again theBA is used for the fields. This example is meant to test how well thedifferential contrast can be detected under the BA in the presence ofclutter. A 3D slice of the final image and its time series are shown inFIGS. 12C and 13 , respectively. The change in contrast with temperatureis clearly visible. The peak contrast is slightly lower than Example 1and more artifacts are present. The artifacts are likely due todecreased accuracy of the BA as well as the fact that the imaging domainencroaches on the near-field of the antennas, where fields have highspatial gradients. The latter effect may be reduced with a largerdiameter cavity. Overall, this shows that a change in temperature may beimaged in the presence of clutter and that the differential measurementsuccessfully subtracted most of the unwanted object scattering.

Example 3: The final example is a simple breast phantom with heatedtarget, as shown in FIGS. 14A-14B. The phantom consists of two plasticbottles creating inner and outer chambers. The inner chamber is filledwith the cavity fluid, meant to mimic the dielectric properties ofglandular breast tissue. The outer chamber is filled with vegetable oilto mimic the lower dielectric properties of fatty breast tissue. A 3Dslice view of the final image and the time series are shown in FIGS. 14Cand 15 . The target and time-varying contrast are clearlydistinguishable. The peak magnitude is slightly lower than that inExamples 1 and 2. Similar to Example 2, the decrease in contrastmagnitude and target blurring is attributed to decreased accuracy of theBA, with the additional possibility that the heat from the object is notconvected from the immediate surroundings of the target as the targetwas in the other examples. Still, the ability of the differentialformulation to successfully subtract the background scattering of thephantom is apparent. It also demonstrates that the BA is actuallysufficient for an object where the BA is otherwise expected to fail.Thus, it is reasonable to conclude that the system, as described herein,has met the requirements for practical thermal monitoring.

To correlate the image with temperature, the pixel intensity at thecenter of the detected target in each example over time may be comparedagainst the temperature curve of the control target in FIG. 9B. Thecurves may be normalized in order to compare the rates of change withtime. Again, complete images are formed every 4 seconds, from which thepixel of peak contrast may be taken. As shown in FIG. 16 , thescattering contrast of all three targets correlates nearly 1:1 withtemperature as function of time. It may be infer from these plots thatthe change in scattering contrast of water is nearly linear withtemperature in the range of hyperthermia temperatures. Furthermore, thisrelation is observed across all of the imaged scenarios (for example,sphere, clutter, phantom, and the like), despite using the BA for thefield estimate.

To determine the temperature resolution, a fourth-order polynomial maybe fit to each of the three pixel intensity curves of FIG. 16 , and thestandard deviations of the residuals may be computed. For Examples 1, 2,and 3, respectively, the standard deviations are σ=0.0062, 0.0045, and0.0163. The temperature resolution is taken as ΔT=2σ|T_(∞)−T₀|. Thetemperature of the target ranges from approximately 55° C. to 22° C.,thus it may be concluded that a temperature change in each case may beresolved to ΔT=0.41, 0.30, and 1.1° C., respectively.

Monitoring and Control

Microwave imaging relies on temperature dependency of the dielectricproperties of tissue rather than on temperature dependency of the protonresonance frequency of water molecules, replacing the need for MRI inthermal therapy monitoring and guidance. The embodiments describedherein include hardware and software features for the successfulimplementation of real-time microwave thermal imaging. Alternatively orin addition, the embodiments described herein may be used to control athermal treatment (for example, a microwave thermal treatment. Thethermal treatment may be controlled automatically or manually by a user(for example, a medical professional).

Whereas prior implementations used the background electric field only,the embodiments described herein include full wave modeling of a thermalprobe in the presence of a segmented anatomy. A robust parser uses ahybrid method (for example, a hybrid of hardware and software) torapidly acquire microwave thermal imaging data. The hardware includes athrough channel connection used as an embedded trigger signaling eachdiscrete measurement within the continuous data stream. The softwareincludes real-time magnitude thresholding to parse a continuous datastream into discrete measurement groups plus real-time median filteringwithin each measurement group to enhance signal-to-noise ratio (SNR).

A custom low-loss coupling medium is implemented to achieve thenecessary SNR to accurately detect small changes in dielectric due totemperature. In some embodiments, the coupling medium is a low lossmaterial with a controlled dielectric constant (for example, ECCOSTOCK®HiK), which has been custom machined to couple microwaves from theimaging cavity to a specific patient anatomy.

An adaptive imaging domain is used to improve accuracy and suppressartifacts. A user-selectable (or machine adaptive) region of interestfor thermal imaging is based on prior knowledge of the treatmentscenario.

Some advantages of the embodiments described herein include: (1) ahigher frame rate possible than MRI monitoring for improved treatmentguidance; (2) a lower cost; (3) a lower system complexity and materialrequirements; (4) a lower space requirement; and (5) a greaterflexibility (for example, no MR compatibility requirement for treatmentsystem).

Real-time microwave thermal monitoring requires knowledge of the tissueproperties in the treatment region as well as any intervening tissues.Currently, this knowledge would need to be obtained using one of thestandard clinical imaging methods (for example, MR, CT, and US).Imperfect knowledge of the tissue properties throughout the imagingvolume may result in artifacts in the thermal image. Sources of errormay include inaccuracies in mapping dielectric tissue data from priorimaging studies, as well as inaccuracies in image segmentation and imageregistration (including those due to patient motion during theprocedure).

FIGS. 17A-17B illustrate a forward model of a 1 cm heated region in ahead. The accompanying table lists permittivity and conductivity valuesat a variety of temperatures. Creating the thermal image of thetreatment region relies upon the temperature dependency of thedielectric properties in the treatment region. With reference to FIGS.18A-18F, various cross-sections illustrating sample electric fields inan unheated condition are illustrated.

With reference to FIGS. 19A-19B, 20A-20B, 21A-21B, and 22A-22B, using apriori knowledge of the electric fields in the unheated head (forexample, operating on the assumption that this information is known fromprior imaging studies), the thermal images are reconstructed in theregion of interest around the treatment location. The temperaturemapping is performed using estimated water content and the knownrelationship between the dielectric properties of water and temperature.The thermal image created using the predicted temperature is compared tothe thermal image created using dielectric scattering in each of FIGS.19A-19B, 20A-20B, 21A-21B, and 22A-22B. FIGS. 19A-19B illustrate thethermal images for T=50° C. and ΔT=13° C., respectively; FIGS. 20A-20Billustrate the thermal images for T=60° C. and ΔT=23° C., respectively;FIGS. 21A-21B illustrate the thermal images for T=70° C. and ΔT=33° C.,respectively; and FIGS. 22A-22B illustrate the thermal images for T=80°C. and ΔT=43° C., respectively.

In further embodiments, the microwave system itself may obtain tissueproperty data directly, either in conjunction with or potentiallyobviating the need for prior imaging studies. Real-time acquisitionmethods may be utilized for the purposes of real-time suppression ofartifacts.

Conclusion

Accordingly, embodiments described herein relate to a real-timemicrowave imaging system to image differential temperature based onchange in dielectric properties of water with the goal of achievingnon-invasive temperature monitoring for thermal therapy systems.

The systems and methods described herein include a cavity-like imagingstructure that is fully modeled in HFSS and coupled with a precomputedlinear inverse scattering solution. The solution is applied in real-timeto a segmented VNA data stream that captures all pairwise scatteredfield measurements. VNA calibration of T/R antenna pairs is acceleratedby using the cavity as an unknown thru standard. The system is tested onwater targets with changing temperature. Changes in scattering contrastdue to temperature were successfully imaged in cluttered environments,including a simple breast phantom.

This study has produced five findings: (1) the dielectric change ofwater over the hyperthermia temperature range (for example, 35° C.-60°C.) is easily detectable with the proposed noninvasive setup at 915 MHz;(2) real-time microwave imaging of differential temperature is possiblewithin the constraints of microwave thermal therapy system parametersusing a VNA; (3) the spatial resolution of thermal imaging is on theorder of the thermal therapy focal spot size in this cavity; (4) 3Dimages of differential temperature can be formed with 1° C. sensitivityat refresh rates of 4 seconds or better; and (5) the rate of change ofscattering contrast magnitude correlates 1:1 with target temperature fora variety of objects.

Section II: Real-Time Microwave Monitoring of Thermal Treatments

In one embodiment, the invention provides a real-time noninvasivethermal monitoring system based on multi-static microwave imaging. Theinversion algorithm, control hardware, and measurement hardware wereoptimized for real-time image formation (refresh rates better than 1frame per second), and the system was validated with heating tests insimple phantoms.

During the process of developing and experimentally validating thesystem, improvements in the flexibility and computational efficiency ofthe forward modeling would b e desirable before real-time thermalimaging would be clinically viable. To address these limitations, anin-house wideband forward solver was developed that integrates (a)object modeling (b) waveport excitation, and (c) S-parameter extractionunder a conformal finite difference time domain (CFDTD) framework. Inparticular, this CFDTD solver has enabled the implementation of a fullyintegrated numerical vector Green's function that addresses the task oflinking the object's scattered field to the measured scattered voltagein a more general and flexible way than the waveport vector Green'sfunction method previously used.

Methods

A. Dual Application of Microwave Imaging to Real-Time Thermal Monitoring

Real-time microwave monitoring of thermal treatments is based upon twofundamental assumptions: knowledge of the complex dielectric propertiesof the tissue in the region of interest prior to heating and knowledgeof how the dielectric properties of each tissue type will change as afunction of temperature. One method of obtaining these dielectricproperties is to infer them from an earlier imaging study (such as MRI).However, this has the potential to introduce error both frominaccuracies in mapping one material property (typically water content)to another (complex permittivity) and from uncertainty in coregisteringthe earlier imaging study with the current patient geometry during thethermal therapy procedure. Instead, it is preferable to generate a mapof the dielectric properties directly at the time of the procedure usingthe same microwave imaging system used for real-time monitoring. It isfor this task, along with the task of continuing to update thedielectric map in real-time during the procedure, that the conformalfinite-difference time domain (CFDTD) forward solver has been developedand integrated into the inversion algorithm shown in flow chart form inFIG. 25 . The iterative inversion algorithm outlined in FIG. 25 uses thefollowing procedure:

-   -   1. In the absence of the object, use the conformal FDTD method        to calculate the incident field inside the object region for        each antenna.    -   2. Use the incident field due to antenna M and the total field        due to antenna N to construct a numerical Greens function        linking the unknown object material property to the scattered        S-parameter SMN in the presence of the imaging cavity and        object. (Note: in the first iteration, the incident field is        substituted for the total field in the object region.)    -   3. Solve for the object material using the local optimization        method with regularization and then use this solution to update        the material properties in the object region.    -   4. With the updated object material, use the conformal FDTD        method to calculate the total field in the object region and the        scattered field in the antenna feed.    -   5. Compare the difference between the measured and        FDTD-calculated scattered fields at the antenna feed. If the        difference is smaller than the threshold, stop, and output        estimated map of complex permittivity. Otherwise, return to step        2 with the updated total field in the object region.

With the volumetric map of the complex permittivity from the inversionalgorithm and the corresponding electric fields from the CFDTD solver inhand, it is then possible to use the Distorted Born Approximation andprecompute a linear inverse scattering solution that can be applied inreal-time to the multi-static scattered S-parameter measurements thatare being continuously acquired by the real-time microwave imaginghardware. This solution, combined with the aforementioned knowledge ofhow complex permittivity varies with temperature for a given tissue typeis what enables real-time thermal monitoring via multi-static arraymicrowave imaging.

B. Conformal FDTD Solver with Waveport Excitation and S-ParameterExtraction

The finite-difference time domain (FDTD) method has proven anincreasingly useful tool in the modeling of electromagnetic problems.Along the way, researchers have made a variety of evolutionaryimprovements designed to address issues of computational efficiency,robustness, and accuracy. Among the most significant of these was thedevelopment of the conformal FDTD (CFDTD) method, which utilizes thefact that the E and H fields are not collocated within one cell andcreates an effective material that captures the geometry of curvilinearobjects without the need for the dense meshing required by thetraditional staircase grid.

In order for the CFDTD method to be practically useful in the forwardmodeling of multi-static antenna array measurements, however, twoadditional modeling challenges were addressed: (1) how to model thesource excitation at the antenna feed, and (2) how to determine thetransmit and receive voltages in the feed to extract the S-parametersrepresenting the actual measured signal. Toward that end, a CFDTDforward solver was developed that integrates (1) a conformal 2Dfinite-difference frequency domain (FDFD) solver to calculate the fielddistribution and propagation constant of the waveport and a total fieldscattered field (TFSF) based soft source to feed that solution into thewaveport, and (2) an S-parameter extraction method compatible with theconformal framework.

C. Integrated Numerical Vector Green's Function

Numerical Green's functions are used in microwave imaging to account forthe effects of the antenna pattern and the presence of the imagingcavity and to link the measured scattered voltage to the object'sscattered field during the inversion process. A waveport vector Green'sfunction proved highly effective in accounting for the variousnon-idealities of real antennas radiating into an imaging cavity, andenabled the development of an experimentally validated imaging system.However, this method relies on inputs from an external forward solver tomodel the electric fields. Since this requires passing the electricfield data from the external solver to the vector Green's implementationin the inversion algorithm, itis inherently cumbersome andcomputationally inefficient. Furthermore, limitations of the forwardsolver, the commercial finite element solver, HFSS) also limited thevector Green's function to the background case (Born Iterative type) orto very simple objects (Distorted Born type). To address theselimitations, a numerical vector Green's function was introduced that isdirectly integrated into the conformal FDTD forward solver discussedabove.

Derivation of Numerical Vector Green's Function from the ReciprocityPrinciple: Using reciprocity, we have:∫_(V) E _(scat)(r)·J _(s)(r)dv=∫ _(V′) E _(inc)(r′)·J_(pol)(r′)dv′  Equation (29)where r is at the antenna feed, r′ is at the object region, E_(scat) isthe scattered field produced by the polarization current J_(pol) andE_(inc) is the incident field produced by the test point current sourceJ_(s). The polarization current produced by the object contrast and thetotal fields in the object region when transmitted from antenna N can berepresented as,J _(pol) ^(N)(r′)=jωϵ ₀(ϵ_(obj)(r′)−ϵ_(b)(r′))E _(obj)^(N)(r′)  Equation (30)

If we want to use the scattered field at antenna M in {circumflex over(p)} direction, then we should also excite a test point source J_(s)^(M)(r) at antenna M in {circumflex over (p)} direction.J _(s) ^(M)(r)=I{circumflex over (p)}δ(r)  Equation (31)

Then, the numerical Green's function to link the scattered field atantenna M when transmitting from antenna N and fora given object'sdielectric property is:

$\begin{matrix}{{E_{scat}^{\hat{p}}(r)} = {\frac{j{\omega\epsilon}_{0}}{I{\delta(r)}}{\int_{V^{\prime}}{\left( {{\epsilon_{obj}\left( r^{\prime} \right)} - {\epsilon_{b}\left( r^{\prime} \right)}} \right){E_{obj}^{N}\left( r^{\prime} \right)}{E_{inc}^{M}\left( r^{\prime} \right)}{dv}^{\prime}}}}} & {{Equation}(32)}\end{matrix}$

FIGS. 27A-27B illustrate two slices of the magnitude of the verticalcomponent of the electric field inside the imaging cavity (shown in FIG.24 ) due to a single transmit antenna at resonance. These fields are inagreement with results obtained previously from both CST MicrowaveStudio and Ansys HFSS, and are currently being used in tests of theinversion algorithm described above and shown in FIG. 25 .

To validate the numerical Green's function, a cubic object withdielectric constant of 16 and size 4×4×4 cm³ was placed in the bottomcenter of the cubic imaging cavity consisting of four panels. Theimaging cavity is filled with a coupling medium with a dielectricconstant of 22. The transmitting antenna is located at the middle of onepanel and the receiving antenna is at the bottom middle of the adjacentpanel. A comparison of the scattered field S-parameters calculated byCST Microwave Studio and those estimated by the numerical Green'sfunction method are shown in FIGS. 28A-28B. As can be seen from FIGS.28A-28B, the scattered fields estimated by the numerical Green'sfunction agree very well with those calculated by CST.

Section III: Conformal Finite-Difference Time Domain Technique

The finite-difference time domain (FDTD) method originally proposed byYee has become a widely used tool in the modeling of electromagneticproblems. However, the original FDTD method with staircase mesh hasdifficulties accurately modeling curved objects. To overcome thisproblem, Dey et al. proposed a conformal FDTD (CFDTD) algorithm with amodified H field update. This method increased the modeling accuracy atthe cost of reduced time-steps. To address this problem, Xiao et al.proposed the enlarged cell technique, while Zagorodnov et al. proposedthe uniformly stable conformal (USC) scheme. However, both of thesemethods are more difficult to implement than the conventional FDTDalgorithm and require additional computational steps. To address this,Benkler et al. introduced the effective material and effective electricfield concept, which implements the CFDTD within the original form ofthe conventional FDTD algorithm. Up to this point, however, no detaileddevelopment of the CFDTD method with waveport excitation and S-parameterextraction has been found in the current literature.

In light of this, a wideband forward solver was developed thatintegrates (a) object modeling (b) waveport excitation, and (c)S-parameter extraction under the conformal FDTD framework. In conformalobject modeling, the effective material method is presented and utilizedto model a coaxial feed and a tapered patch antenna. Compared with theprevious work of modeling coaxial feed with thin wire model andconformal technique, a maximally sparse mesh model for the coaxial feedusing the effective material was proposed. A complementary calibrationscheme is proposed to enhance the impedance modeling accuracy of thecoaxial feed. In terms of waveport excitation, this problem was dividedinto two sub problems, (1) waveport mode calculation, (2) waveportsource feeding. For waveport mode calculation, Zhao et al. introduced acompact 2D finite-difference frequency domain (FDFD) method; however, noconformal development of this method compatible with CFDTD has beenfound in the current literature. Therefore, a conformal 2D FDFDeigenmode solver was developed that solves for the propagation constantand the effective E and H field pattern for waveports with arbitraryshapes. For waveport source feeding, the total field scattered field(TFSF) method was utilized and modified to be compatible with the CFDTDmethod. For S-parameter extraction, Gwarek et al. proposed the generalmethod to extract wideband S-parameters which can be used forinhomogeneous waveguides and different modes. Thus, a modifiedS-parameter extraction scheme was introduced that directly uses theeffective E field in the calculation and can be fully integrated intothe conformal FDTD framework. This implementation of a fully integratedCFDTD solver addresses a wide range of computational electromagneticproblems (including microwave inverse scattering and imagingapplications).

Methods

While offering the advantage of simplicity, staircase meshing suffersfrom the necessity of using a very dense grid in order to accuratelymodel curvilinear objects. To overcome this difficulty, conformalmethods utilize the fact that the E and H fields are not collocatedwithin one Yee cell, creating an effective material that captures thegeometry of curvilinear objects without the need for dense meshing.

In the CFDTD method, the Faraday's Law equation can be discretized withthe non-PEC area of H field and non-PEC length of E field. For example,the discretized updating equation of the Hz field is shown below:

$\begin{matrix}{{H_{z}{❘{\begin{matrix}{n + \frac{1}{2}} \\{i,j,k}\end{matrix} = H_{z}}❘}\begin{matrix}{n - \frac{1}{2}} \\{i,j,k}\end{matrix}} + {\frac{\Delta t}{{\mu \cdot}❘_{i,j,k}} \cdot \left( {{E_{x}{❘{\begin{matrix}n \\{i,{j + 1},k}\end{matrix} \cdot l_{x}}❘}_{i,{j + 1},k}} - {E_{x}{❘{\begin{matrix}n \\{i,j,k}\end{matrix} \cdot l_{x}}❘}_{i,j,k}} - {E_{y}{❘{\begin{matrix}n \\{{i + 1},j,k}\end{matrix} \cdot l_{y}}❘}_{{i + 1},j,k}} + {E_{y}{❘{\begin{matrix}n \\{i,j,k}\end{matrix} \cdot l_{y}}❘}_{i,j,k}}} \right)}} & {{Equation}(33)}\end{matrix}$where {tilde over (S)} represents the non-PEC area of the cell face forH field, l is the non-PEC edge length for the E field as shown in FIGS.29A-29B. The non-PEC area ratio S^(ratio), non-PEC length ratiol^(ratio), effective E field, effective permeability, effectivepermittivity, and effective conductivity as follows:

$\begin{matrix}{S_{z}^{ratio}{❘{i,j,{k = \frac{❘_{i,j,k}}{\Delta{x}_{i}\Delta y_{i}}},{{L_{x}^{ratio}❘_{i,j,k}} = \frac{l_{x}❘_{i,j,k}}{\Delta x_{i}}}}}} & {{Equation}(34)}\end{matrix}$ $\begin{matrix}{\overset{\sim}{E} = {E \cdot L^{ratio}}} & {{Equation}(35)}\end{matrix}$ $\begin{matrix}{{\overset{˜}{\mu} = {\mu \cdot S^{ratio}}},{\overset{\sim}{\epsilon} = \frac{\epsilon}{L^{ratio}}},{\overset{˜}{\sigma} = \frac{\sigma}{L^{ratio}}}} & {{Equation}(36)}\end{matrix}$

Then, by replacing the E field and the material with their effectivevalues, the Faraday's Law and Ampere's Law equations become:

H z ❘ "\[RightBracketingBar]" i , j , k n + 1 2 = H z ❘"\[LeftBracketingBar]" i , j , k n - 1 2 + Δ ⁢ t ⁢ ( ❘"\[LeftBracketingBar]" i , j + 1 , k n - ❘ "\[LeftBracketingBar]" i , j, k n Δ ⁢ y j - ❘ "\[LeftBracketingBar]" i + 1 , j , k n - ❘"\[LeftBracketingBar]" i , j , k n Δ ⁢ x i ) Equation ⁢ ( 5 )$\begin{matrix}{❘_{i,j,k}^{n + 1} = {{{\frac{2 - {\Delta t}}{2 + {\Delta t}} \cdot}❘_{i,j,k}^{n}} + {\left( \frac{2\Delta t}{2 + {\Delta t}} \right) \cdot \left( {\frac{{H_{z}❘_{i,j,k}^{n + \frac{1}{2}}} - {H_{z}❘_{i,{j - 1},k}^{n + \frac{1}{2}}}}{\Delta y_{j}} - \frac{{H_{y}❘_{i,j,k}^{n + \frac{1}{2}}} - {H_{y}❘_{i,j,{k - 1}}^{n + \frac{1}{2}}}}{\Delta z_{k}}} \right)}}} & {{Equation}(6)}\end{matrix}$

As seen above, after substituting E, μ, ϵ, and σ with their effectivevalues, the original FDTD updating equations remain unchanged, makingthis method easy to implement. The application of this conformal FDTDmethod to object modeling and the extension of this method through thedevelopment of compatible waveport excitation and S-parameter extractionschemes is discussed.

A. Conformal Object Modeling

In conventional staircase FDTD modeling, only one type of material isallowed within a given Yee cell. This causes difficulty when modelingcomplex objects with many interfaces, edges, and corners, as theresolution of the objects is therefore limited by the size of the Yeecell. With the conformal FDTD technique, subcell information is utilizedto increase the resolution of object modeling. Subcell information isused to create effective material for the modeling of two frequentlyused objects, a coaxial feed and a patch antenna, and propose animpedance calibration scheme specifically designed to improve coaxialfeed modeling.

-   -   1) Coaxial feed modeling: The modeling of a Teflon coaxial feed        with maximally sparse mesh using effective material is        introduced. The coaxial feed extends in the z direction and its        xy cross section is shown in FIGS. 30A-30B. Based on the Yee        cell shown in FIGS. 29A-29B, we assign the locations of the E        and H fields within each cell as shown in FIGS. 30A-30B. After        using the non-PEC length ratio and non-PEC area ratio based on        the sparse mesh in FIGS. 30A-30B, the effective material value        is calculated and shown in FIGS. 31A-31F.

By utilizing subcell information, the coaxial feed can be modeled withonly four cells containing non-PEC material. Because so few cells areused, there is a small difference between the simulated and theoreticalcharacteristic impedance. While this can be improved with densermeshing, it comes with a higher computation cost; instead, astraightforward calibration scheme to yield a more accurate impedancewas used. The calibration scheme was developed as follows: Define Z_(th)and Z_(sim) as the theoretical and FDTD simulated impedance of thecoaxial feed. U and I are the voltage and current in the coaxial feed,while U′ and I′ are the partial derivative of the voltage and currentalong the direction of the coaxial cable.

$\begin{matrix}{Z_{th} = {\frac{1}{2\pi}\sqrt{\frac{\mu}{\epsilon}}\ln\frac{D}{d}}} & {{Equation}(39)}\end{matrix}$ $\begin{matrix}{{Z_{sim}(\omega)} = \sqrt{\frac{{U(\omega)}{U^{\prime}(\omega)}}{{I(\omega)}{I^{\prime}(\omega)}}}} & {{Equation}(40)}\end{matrix}$ $\begin{matrix}{{\overset{\approx}{\epsilon} = {\overset{\sim}{\epsilon} \cdot \frac{Z_{sim}}{Z_{th}}}},{\overset{\approx}{\mu} = {\overset{˜}{\mu} \cdot \frac{Z_{th}}{Z_{sim}}}}} & {{Equation}(41)}\end{matrix}$

Along with the equations for the theoretical impedance (Equation (39)),Equation (40) is also used to extract the impedance of the coaxial feedfrom FDTD simulation. To calibrate the impedance while retaining thepropagation constant, Equation (41) is used to modify the effectivematerial property. As shown below, the accuracy of the calibratedimpedance is improved to a value very close to the theoreticalimpedance.

-   -   2) Modeling of Tapered Patch Antenna: An experimental inverse        scattering system was developed that uses an imaging cavity with        tapered patch antennas. Thus, it is of interest to utilize the        conformal method to model these antennas as part of an effort to        fully model the entire cavity. In FIGS. 32A-32F, a tapered patch        antenna is shown along with its conformal mesh representation.        As seen from the figure, when the edge of the patch is not        aligned with the Yee grid, the effective material can be used to        accurately capture the non-aligned edges. In addition, the size        of the Yee cell is no longer limited by the thickness of the        metal layer, since both the substrate and the metal material can        be accurately modeled with a single Yee cell and the thickness        of the patch is easily controlled by the effective material        value within the cell. Results and analysis comparing the        simulated and measured reflection coefficient of the tape red        patch antenna are shown below.        B. Waveport Excitation

In FDTD simulations, one of two types of source excitation is typicallyused, lumped element or waveport. While the lumped element source iseasier to implement, it is difficult to excite a specific mode. For awaveport source, the distribution of the E and H fields of a chosen modeis calculated and then fed into the waveguide. Thus, implementing awaveport source involves two problems, how to accurately calculate thedistribution of the E and H fields of a certain mode for an arbitrarilyshaped waveguide and how to feed the calculated mode into the waveguide.Furthermore, the following problems are addressed under the conformalFDTD framework.

-   -   1) Conformal 2D FDFD solver: The calculation of the        cross-sectional field distribution of a waveguide can be        formulated as an eigenvalue problem, which can then be solved        with the FDFD method. Starting from the method developed by Zhao        et al., a non-uniform grid 2D conformal FDFD eigenmode solver is        derived that solves the distribution of the effective electric        field {tilde over (E)} and magnetic field H for a specific mode        in a conformal modeled waveport. Then, the calculated results        from this solver can be directly used in the 3D CFDTD method.        Assuming the waveguide is Z-oriented, the E and H field on XY        plane propagating in the waveguide can be written:        E=[E _(x)(x,y){circumflex over (x)}+E _(y)(x,y)ŷ+E        _(z)(x,y){circumflex over (z)}]e ^(−jβz)  Equation (42a)        H=[H _(x)(x,y){circumflex over (x)}+H _(y)(x,y)ŷ+H        _(z)(x,y){circumflex over (z)}]e ^(−jβz)  Equation (42b)        where β is the propagation constant along the waveguide.        Utilizing the conformal method with effective material and        effective electric field, the frequency domain Maxwell equations        can be written in the forms:

$\begin{matrix}{{{\nabla \times \overset{˜}{E}} = {{- j}K_{0}\eta_{0}H}}} & {{Equation}\left( {43a} \right)}\end{matrix}$ $\begin{matrix}{{\nabla \times H} = {jK_{0}\frac{1}{\eta_{0}}\overset{˜}{E}}} & {{Equation}\left( {43b} \right)}\end{matrix}$where K₀ is the free space wavenumber, η₀ is the free space waveimpedance. By substituting Equation (42) into Equation (43) and takingthe difference form of the equations, we can get the expressions of Hfield in terms of {tilde over (E)} field, and vice versa. For example,the H_(x) field can be expressed as:

$\begin{matrix}{{H_{x}❘_{i,j}} = {{\frac{j}{K_{0}\eta_{0}{❘_{i,j} \cdot {dy}_{e}}❘_{j + 1}}\left\lbrack \left. {❘_{i,{j + 1}} -} \right|_{i,j} \right\rbrack} - {e^{\frac{{- j}\beta\Delta z}{2}}\frac{\beta}{k_{0}\eta_{0}❘_{i,j}}❘_{i,j}}}} & {{Equation}(44)}\end{matrix}$

Similarly, the expressions of H_(y), H_(z), {tilde over (E)}_(x), {tildeover (E)}_(y), {tilde over (E)}_(z) are acquired. Approaching Δz to be0, reducing the 3D Yee mesh to 2D grid, and eliminating the {tilde over(E)}_(z) and H_(z) from the derived six equations, the final formulationof the conformal 2D FDFD solver is as follows:

$\begin{matrix}{{\beta K_{0}H_{x}❘_{i,j}} = {{\alpha_{1}^{hx}\left( {❘_{{i - 1},j} - ❘_{{i - 1},{j + 1}}} \right)} - {\alpha_{2}^{hx}\left( {❘_{i,j} - ❘_{i,{j + 1}}} \right)} - {\alpha_{3}^{hx}❘_{{i - 1},j}} + {\alpha_{4}^{hx}❘_{i,j}} - {\alpha_{5}^{hx}❘_{{i + 1},j}}}} & {{Equation}(45)}\end{matrix}$ $\begin{matrix}{{\beta K_{0}H_{y}❘_{i,j}} = {{\alpha_{1}^{hy}❘_{i,{j - 1}}} + {\alpha_{2}^{hy}❘_{i,j}} + {\alpha_{3}^{hy}❘_{i,{j + 1}}} - {\alpha_{4}^{hy}\left( {❘_{i,j} - ❘_{{i + 1},{j - 1}}} \right)} - {\alpha_{5}^{hy}\left( {❘_{i,j} - ❘_{{i + 1},j}} \right)}}} & {{Equation}(46)}\end{matrix}$ $\begin{matrix}{{\beta K_{0}❘_{i,j}} = {{\alpha_{1}^{ex}\left( {{H_{x}❘_{i,j}} - {H_{x}❘_{i,{j - 1}}}} \right)} + {\alpha_{2}^{ex}\left( {{H_{x}❘_{{i + 1},{j - 1}}} - {H_{x}❘_{{i + 1},j}}} \right)} + {\alpha_{3}^{ex}H_{y}❘_{{i - 1},j}} + {\alpha_{4}^{ex}H_{y}❘_{i,j}} + {\alpha_{5}^{ex}H_{y}❘_{{i + 1},j}}}} & {{Equation}(47)}\end{matrix}$ $\begin{matrix}{{\beta K_{0}❘_{i,j}} = {{{- \alpha_{1}^{ey}}H_{x}❘_{i,{j - 1}}} + {\alpha_{2}^{ey}H_{x}❘_{i,j}} - {\alpha_{3}^{ey}H_{x}❘_{i,{j + 1}}} + {\alpha_{4}^{ey}\left( {{H_{y}❘_{{i - 1},j}} - {H_{y}❘_{i,j}}} \right)} - {\alpha_{5}^{ey}\left( {{H_{y}❘_{{i - 1},{j + 1}}} - {H_{y}❘_{i,{j + 1}}}} \right)}}} & {{Equation}(48)}\end{matrix}$where, in the coefficients α, dx_(e|i) is the distance between the(i)^(th) and the (i+1)^(th) electric field mesh grid in x direction,dy_(h|j) is the distance between the (j−1)^(th) and (j)^(th) magneticfield mesh grid in y direction. The expressions of α^(hx) are shownbelow (sample coefficients of the 2D FDFD solver).

$\begin{matrix}{\alpha_{1}^{hx} = \frac{1}{\eta_{0}{❘_{{i - 1},j} \cdot {dx}_{h}}{❘_{i} \cdot {dy}_{e}}❘_{j + 1}}} & {{Equation}\left( {49a} \right)}\end{matrix}$ $\begin{matrix}{\alpha_{2}^{hx} = \frac{1}{\eta_{0}{❘_{i,j} \cdot {dx}_{h}}{❘_{i} \cdot {dy}_{e}}❘_{j + 1}}} & {{Equation}\left( {49b} \right)}\end{matrix}$ $\begin{matrix}{\alpha_{3}^{hx} = \frac{1}{\eta_{0}{❘_{{i - 1},j} \cdot {dx}_{e}}{❘_{i} \cdot {dx}_{h}}❘_{i}}} & {{Equation}\left( {49c} \right)}\end{matrix}$ $\begin{matrix}{\alpha_{4}^{hx} = {\frac{{{- K_{0}^{2}} \cdot}❘_{i,j}}{\eta_{0}} + \frac{1}{\eta_{0}{❘_{i,j} \cdot {dx}_{e}}{❘_{i + 1} \cdot {dx}_{h}}❘_{i}} + \frac{1}{\eta_{0}{❘_{{i - 1},j} \cdot {dx}_{e}}{❘_{i} \cdot {dx}_{h}}❘_{i}}}} & {{Equation}\left( {49d} \right)}\end{matrix}$ $\begin{matrix}{\alpha_{5}^{hx} = \frac{1}{\eta_{0}{❘_{i,j} \cdot {dx}_{e}}❘_{i + 1}{dx}_{h}❘_{i}}} & {{Equation}\left( {49e} \right)}\end{matrix}$Other coefficients are similar. The four equations above are used toformulate an eigenvalue problem, where the {tilde over (E)} and H fieldsare the eigenvectors and βK₀ is the eigenvalue. The problem isreformulated as the generalized eigenvalue problem below:[A]·{x}=λ[B]·{x}  Equation (50)where B is a unit matrix. The zero field within the PEC region isrealized in two ways. For {tilde over (E)}_(z) and H_(z) field withinthe PEC region, a large number is assigned to {tilde over (ε)}_(z) and{tilde over (μ)}_(z) to ensure the corresponding field term in theequation becomes zero. To set the {tilde over (E)}_(x), {tilde over(E)}_(y),H_(x), H_(y) within the PEC region to 0, the correspondingdiagonal terms of matrix A and B are multiplied by a very large number.Usually, 10¹⁵ is sufficient. Then, the ith equation will become10¹⁵·(a_(ii)−λb_(ii))x_(i)=0, which ensures x_(i)=0. Each solvedeigenvector represents the distribution of the E and H field of certainmode within a waveport. The total power of a specific mode within thewaveport can be calculated by integrating the mode field E and H overthe non-PEC area {tilde over (S)} of the waveport. However, as shown inEquation (51), through simple transformation, it is equivalent todirectly integrating the {tilde over (E)} and H over the total area S ofthe waveport, which is the surface area of related Yee cells.P=½∫∫(E×H*)·d{tilde over (S)}=½∫∫({tilde over (E)}×H*)·dS  Equation (51)

-   -   2) Source Feeding: To accurately feed the mode fields into the        waveport, we use a modified TFSF soft-source excitation method        that can be directly used in the CFDTD algorithm and can        accurately excite the desired fields with no retro-reflection        and without being affected by the Courant condition number. The        original TFSF method was proposed to excite plane waves and to        divide the computation domain into a total field region and a        scattered field region. To apply the TFSF method to waveport        excitation, the electric source field is fed into the waveport        during the magnetic field updating loop and vice versa. For        example, the effective electric field source {tilde over        (E)}^(s) is fed into the waveport through:

H x n + 1 2 = H x n - 1 2 - ( Δ ⁢ t ) ⁢ ( [ ∇ × E ˜ ] x n - · f ⁡ ( t ) Δ ⁢z ) Equation ⁢ ( 52 ⁢ a ) H y n + 1 2 = H y n - 1 2 - ( Δ ⁢ t ) ⁢ ( [ ∇ × E˜ ] y n + · f ⁡ ( t ) Δ ⁢ z ) Equation ⁢ ( 52 ⁢ b )where Δz is the Yee cell length in z direction at the source point,{tilde over (E)}_(x) ^(s), {tilde over (E)}_(y) ^(s) are the sourcefield coefficients calculated by the conformal 2D FDFD solver and f(t)is the time domain function of the signal. Here, the effective materialand effective E field are used to make this formulation compatible withthe conformal FDTD technique. The magnetic field source H^(s) can also be fed into the waveport using similar schemes. Feeding either {tildeover (E)}^(s) or H^(s) will excite a bi-directional propagating wave,which can be used to accurately extract S-parameters of various waveportmodes with the method described below. In the TEM mode case, one canexcite a unidirectional propagating wave by feeding both {tilde over(E)}^(s) and H^(s) simultaneously into the waveport with the TFSFmethod.

C. S-Parameter Extraction

For S-parameter extraction, the method starts with Gwarek et al. and ismodified to be compatible with the conformal technique. As shown in FIG.33 , the waveport with the complex frequency-shifted perfect matchedlayer (CPML) absorbing boundary condition is terminated. Either theincident {tilde over (E)} or the H source is fed on the source planewhich excites a bi-directional propagating wave. The total {tilde over(E)} and H field are recorded in the sensor plane. The sensor plane isplaced between the source plane and the device under test (DUT). Toextract the S-parameters from waveport, the voltage and current of aparticular mode within the waveport need to be defined. The definitionof waveport voltage and current from Gwarek et al. is used, and with thesame simple transformation as Equation (51), they are modified to becompatible with the CFDTD method as follows:U(t)=∫∫_(s) {tilde over (E)}(i,j,k,t)×h(i,j,ω)·dS  Equation (53a)I(t)=∫∫_(S) {tilde over (e)}(i,j,ω)×H(i,j,k,t)·dS  Equation (53b)where {tilde over (e)} and h are normalized mode fields calculated bythe 2D conformal FDFD solver, and {tilde over (E)} and H are the fieldscomputed by the CFDTD at the sensor plane. S is the total area of thewaveport including the PEC area. In this way, the {tilde over (e)} and{tilde over (E)} field from the conformal solvers can be directly used.Meanwhile, in contrast to integrating over the non-PEC area, integratingover the total area S allows us to directly use the surface area ofrelated Yee cell which is readily available. From Gwarek et al., theincident wave a(ω) and reflected wave b(ω) are defined as:

$\begin{matrix}{{a(\omega)} = \frac{{U(\omega)} + {{Z(\omega)}{I(\omega)}}}{\sqrt[2]{Z(\omega)}}} & {{Equation}\left( {54a} \right)}\end{matrix}$ $\begin{matrix}{{b(\omega)} = \frac{{U(\omega)} - {{Z(\omega)}{I(\omega)}}}{\sqrt[2]{Z(\omega)}}} & {{Equation}\left( {54b} \right)}\end{matrix}$where Z(ω) is the impedance of the waveport calculated with Equation(40). S-parameter of the device under test can then be extracted whendividing b(ω) by the corresponding a(ω).

Results

In this section, the conformal object modeling is validated, themodified TFSF feeding method, and the simple coaxial calibration schemewith a Teflon coaxial cable. Then, the conformal 2D FDFD solver with arectangular waveguide and a coaxial feed is validated. Finally, theconformal modeling, waveport excitation and S-parameter extraction witha tapered patch antenna and a waveguide with two coaxial probes arecollectively validated.

A. Coaxial Feed Modeling Validation

In this example, the conformal FDTD modeling and simulation results of aTeflon filled coaxial feed using effective material with (1) a sparsemesh, (2) a dense mesh and (3) a modified sparse mesh with calibrationis demonstrated. The coaxial feed has the same dimension as in FIGS.30A-30B. A modulated Gaussian pulse with a bandwidth of 200 MHz to 6 GHzis fed into the coaxial cable with the modified TFSF method. Thetimestep used in the simulation is 2 ps, and the voltage waveform intime is recorded at both the source plane and the observation plane 8 cmaway. In all three meshing cases, simulation results indicated a shiftof 192 timesteps (0.384 ns) between the source signal and the signal inthe observation plane. Thus, the wave propagation speed in the simulatedcable is 2.08×10⁸ m/s—very close to the theoretical speed of 2.07×10⁸m/s. Also, FIGS. 34A-34B shows that the observed voltage waveform iswell preserved for all three cases. Finally, the impedance of the cableis calculated with Equation (40). As shown in FIGS. 34A-34B, withincreased mesh density, the simulated impedance approaches thetheoretical value. However, with calibration, the modified sparse meshyields the most accurate impedance.

B. Conformal 2D FDFD Mode Solver Validation

In this section, the 2D conformal FDFD mode solver will be evaluatedwith the calculation of the mode and field distribution of a rectangularwaveguide and a Teflon filled coaxial feed.

-   -   1) Rectangular waveguide: The dimensions of the air-filled        rectangular waveguide are 19.05×9.525 mm². As shown in FIG. 35 ,        the calculated propagation constant (first five modes shown) are        in good agreement with their theoretical value.    -   2) Coaxial feed: The 2D conformal FDFD solver can also be used        to solve for the coaxial feed in FIGS. 30A-30B with curved        edges. The calculated propagation constant matches very well        with the theoretical value across the band. The field        distribution of the TEM mode shown in FIGS. 36A-36D also matches        very well with theory. Thus, we have validated that this        conformal FDFD solver is very accurate when solving waveports        with curvilinear edges.

C. End-to-End Validation with Patch Antenna Reflection CoefficientModeling

In this example, we model the reflection coefficient of a probe fed,tapered patch antenna (FIGS. 32A-32F), developed for medicalapplications. This was selected as a practical example that collectivelytests each of the methods developed herein: conformal object modeling(sparse mesh coax feed modeling with impedance calibration), waveportexcitation, and S-parameter extraction. A comparison with experimentalresults in FIGS. 37A-37B show good agreement in both the magnitude andphase of S₁₁, and serves as an end-to-end validation of all of theconformal methods developed above working in concert.

D. End-to-End Validation with Transmission Coefficient Modeling in aWaveguide

Lastly, the modeling of the transmission S parameter between twoarbitrarily placed coaxial probes in a WR-187 waveguide (as shown inFIGS. 38A-38B) is validated. The waveguide is hollow brass with outerdimensions of 2.54×5.08×15.2 cm³ and open on either end. One probe islocated 1.91 cm inward from one corner along both axes and the other islocated 2.54 cm and 1.27 cm from the other corner. A dense mesh is usedto model the coaxial probes, and the TEM field distribution from FIGS.36A-36D is used to excite the coaxial feed. The forward transmissionS-parameters extracted from the simulation and measured with the networkanalyzer presented below indicate good agreement (as shown in FIGS.39A-39B) and serve as final validation of the methods developed herein.

Conclusion

Integrated object modeling, waveport excitation, and S-parameterextraction under the conformal FDTD framework was investigated. First,conformal modeling with effective material method is discussed, and thenis utilized to model the coaxial feed with a minimum number of Yeecells. A complementary calibration scheme is proposed to increase theimpedance modeling accuracy of the coaxial feed, while keeping thepropagation constant unchanged. Second, a conformal 2D FDFD solver isdeveloped which is capable of accurately computing the propagationconstant and mode field distribution of a waveport with curved edges. Inaddition, the solver directly computes the {tilde over (E)} field whichcan then be naturally fed into the CFDTD algorithm. A modified TFSF softsource excitation method compatible with the CFDTD algorithm is alsodeveloped, and its accuracy is validated by comparing the source signaland excited signal within a coax feed. Third, a modified S-parameterextraction scheme is introduced which directly utilizes the {tilde over(E)} field. This modified method allows integrating over the total areaof the waveport when computing mode voltage and current without the needto distinguish between PEC areas and non-PEC areas. Last, all of themethods above are fully integrated into a complete forward solver andvalidated by several simulations and experiments.

Section IV: Numerical Vector Green's Function for S-ParameterMeasurement with Waveport Excitation

In solving microwave inverse scattering problems, the unknown objectmaterial property is often related to the scattered electric fieldthrough the volume integral equation (VIE). In the case of a homogeneousbackground, the kernel of the VIE is a dyadic Green's function, that canbe obtained analytically. However, in inverse scattering experiment,antennas or probes are always present in the background, making thebackground inhomogeneous. In this case, there is no analytical form ofthe Green's function, and the kernel of the VIE can only b e obtainednumerically with reciprocity principle. Moreover, the measurement ofchoice in microwave inverse scattering experiment is always a set ofscattered S-parameter with the vector network analyzer (VNA). TheS-parameter, defined as the voltage ratio, is a scalar instead of avector. Thus, a new vector Green's function kernel within the VIE isneeded which relates the total field within the object domain to thescattered S-parameter (scalar) instead of scattered field (vector). Inaddition, to calculate the vector Green's function numerically, we needto excite antenna feeds. Instead of using a dipole or lumped elementsource, a more accurate waveport source, which could excite the desiredmodes of the feed, is used here. A numerically calculated vector Green'sfunction under waveport excitation is described that serves as a newkernel in the VIE to link the object material property to the measuredscattered S-parameter.

Previously, for inverse scattering with a homogeneous background, ananalytical dyadic Green's function is used as the kernel of the VIE.With an inhomogeneous background, dipole source, plane wave source, orantenna gain patterns are used to in the inverse scattering experiment,which is not sufficiently accurate to model the antennas and link thefields at the antenna feed to the voltage. Hagness et al. uses aresistive voltage source which could not account for the different modesin the antenna feed. Yuan et al. uses a waveport source to formulate theGreen's function, however, the output is in the form of scatteredvoltage and it did not account for the case that transmitting andreceiving waveports have different characteristic s and impedance. Asource scattering matrix approach is used to model the antenna,formulate the Green's function and links the object property to theS-parameter. However, this approach is not fully numerical which havedifficulties accounting for the inhomogeneous background other than theantennas. The full numerical vector Green's function could directlylinks the object property to the S-parameter in an inhomogeneousbackground with waveport excitation, where transmitting and receivingwaveports could have different characteristics, impedance and modes.

The general form of the new vector Green's function for S-parametermeasurement with waveport source is derived. To compute the vectorGreen's function, the incident field and total field within the objectdomain are obtained numerically with the FDTD method under waveportexcitation. Note, this general form of the new vector Green's functionis not tied with the FDTD method, other numerical methods (e.g. FEM,FDFD) can also be used. The rest of the paper is organized as follows:In section II, the detailed derivation of the new vector Green'sfunction is given and the validation steps are presented. Below, threeexamples are given to validate the vector Green's function. The firstexample validates the vector Green's function in simulation, when thetransmitting and receiving waveports have different geometry andcharacteristic impedance. The second and third example validate thevector Green's function with measurements, where the coaxial feedwaveport is used. The third example using the imaging cavity serves as aprecursor for our inverse scattering experiment with this vector Green'skernel in the VIE.

Derivation

In this section, the volume integral equation is introduced to formulatethe inverse scattering problem. Then, the Green's function to link theobject property to the scattered field for homogeneous background andinhomogeneous background is presented. Next, the waveport source andS-parameter extraction will be introduced under the FDTD framework, andnumerical vector Green's function for S-parameter measurement withwaveport excitation is demonstrated and derived. Last, the validationsteps for the new numerical vector Green's function are presented.

A. Volume Integral Equation

In inverse scattering problems, the volume integral equation is oftenused to relate the unknown objects' dielectric property to the measuredscattered fields.

$\begin{matrix}{{E(r)} = {{E_{inc}(r)} + {k_{b}^{2}{\int{\int{\int{{{\overset{¯}{G}\left( {r,r^{\prime}} \right)} \cdot \left( {{{\Delta\epsilon}\left( r^{\prime} \right)} + \frac{\Delta{\sigma\left( r^{\prime} \right)}}{j{\omega\epsilon}_{b}}} \right)}{E\left( r^{\prime} \right)}{dV}^{\prime}}}}}}}} & {{Equation}(55)}\end{matrix}$where r is the observation position vector, r′ is the object domainposition vector. E is the total field and E_(inc) is the incident field.k_(b) is the lossless background wavenumber with k_(b) ²=ω²μ₀ε_(b).ϵ_(b) is the background permittivity with ϵ_(b)=ϵ₀ϵ_(rb), where ϵ_(rb)is the background relative permittivity. The object's material contrastfunction is defined as

$\begin{matrix}{{{\Delta\epsilon}(r)} = \frac{{\epsilon(r)} - {\epsilon}_{b}}{\epsilon_{b}}} & {{Equation}\left( {56a} \right)}\end{matrix}$ $\begin{matrix}{{{\Delta\sigma}(r)} = {{\sigma(r)} - \sigma_{b}}} & {{Equation}\left( {56b} \right)}\end{matrix}$where ϵ(r) and σ(r) are the permittivity and conductivity of the object.Δϵ(r) is the normalized permittivity contrast function which isunitless. Δσ(r) is the conductivity contrast function which has the unitof Siemens per meter. G(r, r′) is the dyadic Green's function. In ahomogeneous background medium, the dyadic Green's function can becomputed analyticallyas

$\begin{matrix}{{\overset{¯}{G}\left( {r,r^{\prime}} \right)} = {\left\lbrack {\overset{¯}{I} + \frac{\nabla^{\prime}\nabla^{\prime}}{k^{2}}} \right\rbrack{g\left( {r,r^{\prime}} \right)}}} & {{Equation}(57)}\end{matrix}$where

$\begin{matrix}{{g\left( {r,r^{\prime}} \right)} = \frac{e^{{- {jk}}{❘{r - r^{\prime}}❘}}}{4\pi{❘{r - r^{\prime}}❘}}} & {{Equation}(58)}\end{matrix}$with k as the background wavenumber given by

$\begin{matrix}{k^{2} = {k_{b}^{2}\left( {1 + \frac{\sigma_{b}}{j{\omega\epsilon}_{b}}} \right)}} & {{Equation}(59)}\end{matrix}$

The scattered field can be written as the difference between the totalfield and the incident fieldE _(scat)(r)=E(r)−E _(inc)(r)  Equation (60)and if the object function O(r′) is defined as

$\begin{matrix}{{O\left( r^{\prime} \right)} = {{{\Delta\epsilon}\left( r^{\prime} \right)} + \frac{\Delta{\sigma\left( r^{\prime} \right)}}{j{\omega\epsilon}_{b}}}} & {{Equation}(61)}\end{matrix}$the volume integral equation then becomesE _(scat)(r)=k _(b) ² ∫∫∫O(r′) G (r,r′)·E(r′)dV′  Equation (62)

B. Green's Function for Inhomogeneous Background with Dipole Source

When the background medium is inhomogeneous, there is often noanalytical form of the Green's function. This is often the case inmicrowave imaging experiments with the presence of antennas and theimaging cavity in the background. When illuminating an object atlocation r′, the total electric field within the object region producesa polarization current which can be written as

$\begin{matrix}{{J_{pol}\left( r^{\prime} \right)} = {j{{\omega\epsilon}_{b}\left( {{{\Delta\epsilon}\left( r^{\prime} \right)} + \frac{\Delta{\sigma\left( r^{\prime} \right)}}{j{\omega\epsilon}_{b}}} \right)}{E\left( r^{\prime} \right)}}} & {{Equation}(63)}\end{matrix}$

Using reciprocity, if the scattered field produced by the polarizationcurrent at the receiver's location r is defined as E_(scat)(r), removethe object from the scenario and excite a current source J_(s) at thereceiver's location r, and record the incident field E_(inc)(r′) withinthe objects' region r′, the result is∫∫∫E _(scat)(r)·J _(s)(r)dV=∫∫∫E _(inc)(r′)·J _(pol)(r′)dV  Equation(64)If J_(s) is a dipole source pointing at {circumflex over (p)} directionJ _(s)(r)=Idlδ(r−r _(p)){circumflex over (p)}  Equation (65)then the scattered field can be written as

$\begin{matrix}{{{E_{scat}(r)} \cdot \overset{\hat{}}{p}} = {\frac{\int{\int{\int{{{E_{inc}\left( r^{\prime} \right)} \cdot {J_{pol}\left( r^{\prime} \right)}}{dV}^{\prime}}}}}{Idl} = {k_{b}^{2}{\int{\int{\int\frac{{O\left( r^{\prime} \right)}{{E_{inc}\left( r^{\prime} \right)} \cdot {E\left( r^{\prime} \right)}}{dV}^{\prime}}{{- j}\omega\mu_{0}{Idl}}}}}}}} & {{Equation}(66)}\end{matrix}$Thus, the vector numerical Green's function for the inhomogeneousbackground with a dipole source is

$\begin{matrix}{{G\left( {r,r^{\prime}} \right)} = \frac{E_{inc}\left( r^{\prime} \right)}{{- j}\omega\mu_{0}{Idl}}} & {{Equation}(67)}\end{matrix}$

This function is termed a numerical vector Green's function, because:(1) E_(inc) is computed numerically, and (2) the Green's function is nolonger a dyad but a vector. The Green's function here can be computedwith various numerical electromagnetic methods (for example, FEM, FDFD,and FDTD). Below, the FDTD method is utilized; however, the formulationof the numerical vector Green's function for S-parameter measurementwith waveport excitation is not specific to any numerical methods.

C. Waveport Excitation and S-Parameter Extraction

In the simulation of realistic microwave systems, the waveport source isoften used to excite the feed rather than a simple dipole source, aswaveport source could accurately excite various waveguides with thedesired modes. To implement the waveport source with FDTD method,calculate first the field distribution of E and H field of the desiredmode with the 2D FDFD solver on the waveport surface. The solved fieldmode template for electric and magnetic field are e^(t)(r) and h^(t)(r),which are then normalized to produce unit power.½∫∫_(A) e ^(t)(r)×h ^(t)(r)·dA=1  Equation (68)where A is the non-PEC surface area of the waveport. If the sourcesignal waveform is f(t), the waveport is excited in FDTD simulation withthe TFSF method shown below

$\begin{matrix}{H_{x}^{n + \frac{1}{2}} = {H_{x}^{n - \frac{1}{2}} - {\left( \frac{\Delta t}{\mu_{x}} \right)\left( {\left\lbrack {\nabla{+ E}} \right\rbrack_{x}^{n} - \frac{e_{y}^{t} \cdot f^{n}}{\Delta z}} \right)}}} & {{Equation}\left( {69a} \right)}\end{matrix}$ $\begin{matrix}{H_{y}^{n + \frac{1}{2}} = {H_{y}^{n - \frac{1}{2}} - {\left( \frac{\Delta t}{\mu_{y}} \right)\left( {\left\lbrack {\nabla \times E} \right\rbrack_{y}^{n} + \frac{e_{x}^{t} \cdot f^{n}}{\Delta z}} \right)}}} & {{Equation}\left( {69b} \right)}\end{matrix}$ $\begin{matrix}{E_{x}^{n + 1} = {E_{x}^{n} + {\left( \frac{\Delta t}{\epsilon_{x}} \right)\left( {\left\lbrack {\nabla \times H} \right\rbrack_{x}^{n + \frac{1}{2}} + \frac{h_{y}^{t} \cdot f^{n + \frac{1}{2}}}{\Delta z}} \right)}}} & {{Equation}\left( {69c} \right)}\end{matrix}$ $\begin{matrix}{E_{y}^{n + 1} = {E_{y}^{n} + {\left( \frac{\Delta t}{\epsilon_{y}} \right)\left( {\left\lbrack {\nabla \times H} \right\rbrack_{y}^{n + \frac{1}{2}} - \frac{h_{x}^{t} \cdot f^{n + \frac{1}{2}}}{\Delta z}} \right)}}} & {{Equation}\left( {69d} \right)}\end{matrix}$

Note, both E and H source field are excited at the waveport to createthe unidirectional propagation of the wave from the waveport. With theTFSF method, it is known explicitly that the incident electric field atthe waveport will be exactly e(r)f(t).

In microwave measurement with VNA, the S-parameter between waveport mwith the i^(th) mode and waveport n with the j^(th) mode can be definedas

$\begin{matrix}{{S_{m.n}^{i,j}(\omega)} = \frac{{V_{m}^{i -}(\omega)}\sqrt{Z_{n}^{j}(\omega)}}{{V_{n}^{j +}(\omega)}\sqrt{Z_{m}^{i}(\omega)}}} & {{Equation}(70)}\end{matrix}$where V_(m) ^(i−) is the received voltage of the i^(th) mode forwaveport m, V_(n) ^(j+) is the incident voltage of the j^(th) mode forwaveport n. Z_(m) ^(i) and Z_(n) ^(j) are the i^(th) and j^(th) modeimpedance for waveport m and n. The voltage and current of the i^(th)mode for waveport m can be defined as

$\begin{matrix}{{V_{m}^{i}(\omega)} = {\int{\int_{A_{m}}{{E\left( {r,\omega} \right)} \times {{{\overset{\_}{h}}_{m}^{ti}(r)} \cdot {dA}}}}}} & {{Equation}\left( {71a} \right)}\end{matrix}$ $\begin{matrix}{{I_{m}^{i}(\omega)} = {\int{\int_{A_{m}}{{H\left( {r,\omega} \right)} \times {{{\overset{\_}{e}}_{m}^{ti}(r)} \cdot {dA}}}}}} & {{Equation}\left( {71b} \right)}\end{matrix}$where E and H are the electric and magnetic field at waveport m. A_(m)is the non-PEC surface area for waveport m. ē_(m) ^(ti) n and h _(m)^(ti) are the normalized i^(th) mode template of the electric andmagnetic field for waveport m, which shall satisfy

$\begin{matrix}{{\int{\int_{A_{m}}{{{{\overset{\_}{e}}_{m}^{ti}(r)} \cdot {{\overset{\_}{e}}_{m}^{ti}(r)}}{dA}}}} = 1} & {{Equation}\left( {72a} \right)}\end{matrix}$ $\begin{matrix}{{\int{\int_{A_{m}}{{\overset{\_}{h}}_{m}^{ti}{(r) \cdot {\overset{\_}{h}}_{m}^{ti}}(r){dA}}}} = 1} & {{Equation}\left( {72b} \right)}\end{matrix}$The waveport mode voltage and current defined here represent theamplitudes of the electric and the magnetic field, and is only analogousto the traditional voltage and current concept for the transmission linewith TEM mode. The impedance of the waveport is also computednumerically with the differential method in Gwarek et al. The impedanceof the i^(th) mode for waveport m can be written as

$\begin{matrix}{{Z_{m}(\omega)} = \sqrt{\frac{{V_{m}^{i}(\omega)}{V_{m}^{i^{\prime}}(\omega)}}{{I_{m}^{i}(\omega)}{I_{m}^{i^{\prime}}(\omega)}}}} & {{Equation}(73)}\end{matrix}$

D. Numerical Green's Kernel for S-Parameter Measurement with WaveportSource

In microwave imaging experiments, the scattered S-parameter is notmeasured directly. Instead, the S-parameter with and without thepresence of the object are measured. The difference between the two isthe scattered S-parameter:S _(m,n) ^(scat)(ω)=S _(m,n) ^(obj)(ω)−S _(m,n) ^(inc)(ω)  Equation (74)

Using this definition of the scattered S-parameter, we are interested inbuilding a new volume integral equation that directly links unknownobject material property to the measurement, which can be written asS _(m,n) ^(scat)(ω)=k _(b) ² ∫∫∫O(r′)G _(m,n)(r′,ω)·E_(n)(r′,ω)dV′  Equation (75)where E_(n)(r′, ω) is the total field inside the object region due tothe excitation of waveport n, and G_(m,n)(r′, ω) is the new numericalvector Green's function which links the object material property to thescattered S-parameter between waveport m and n.

To derive the new form of the Green's function, start from reciprocity∫∫∫_(V) _(m) E _(scat)(r,ω)·J _(s)(r,ω)dV=∫∫∫ _(V) _(obj) E_(inc)(r′,ω)·J _(pol)(r′,ω)dV′  Equation (76)where V_(obj) is the volume of the object region, and V_(m) is thevolume of the waveport which equals to A_(m)Δ_(z), the non-PEC surfacearea of the waveport times the mesh size in the propagation direction.In order to link to the S-parameter, E_(scat) on the LHS of the equationto the scattered voltage is transformed. In FDTD, the current sourceJ_(s) on the LHS of the Equation (76) is fed into the waveport through

$\begin{matrix}{E_{x}^{n + 1} = {E_{x}^{n} + {\left( \frac{\Delta t}{\epsilon_{x}} \right)\left( {\left\lbrack {\nabla \times H} \right\rbrack_{x}^{n + \frac{1}{2}} - J_{y}^{n + \frac{1}{2}}} \right)}}} & {{Equation}\left( {77a} \right)}\end{matrix}$ $\begin{matrix}{E_{y}^{n + 1} = {E_{y}^{n} + {\left( \frac{\Delta t}{\epsilon_{y}} \right)\left( {\left\lbrack {\nabla \times H} \right\rbrack_{y}^{n + \frac{1}{2}} - J_{x}^{n + \frac{1}{2}}} \right)}}} & {{Equation}\left( {77b} \right)}\end{matrix}$

If J_(s) takes the form

$\begin{matrix}{{J_{s}\left( {r,t} \right)} = \frac{\overset{\hat{}}{n} \times {{\overset{¯}{h}}_{m}^{ti}(r)}{f(t)}}{\Delta z}} & {{Equation}(78)}\end{matrix}$take the Fourier transform of J_(s) and substitute it into the LHS ofEquation (76). Then Equation (76) becomes

$\begin{matrix}{{\int{\int_{A_{m}}{{E_{scat}\left( {r,\omega} \right)} \times {{\overset{¯}{h}}_{m}^{ti}(r)}{{f(\omega)} \cdot {dA}}}}} = {\int{\int{\int_{V_{obj}}{{{E_{inc}\left( {r^{\prime},\omega} \right)} \cdot {J_{pol}\left( {r^{\prime},\omega} \right)}}{dV}^{\prime}}}}}} & {{Equation}(79)}\end{matrix}$where the LHS of the equation collapses to a surface integral. As shownabove, with the proper choice of the current source J_(s), the LHS ofthe equation becomes the scattered voltage.

Assume the polarization current is due to the waveport excitation atwaveport n. If waveport n is excited with the i^(th) mode field templatee_(n) ^(tj)(r) and h_(n) ^(tj)(r) using the TFSF method. The same signalwaveform f(t) as Equation (78) is used. Then the incident electric fieldat waveport n can be written asE _(n) ^(j)(r,t)=e _(n) ^(tj)(r)f(t)  Equation (80)Take the Fourier transform of the incident the electric field andnormalize the magnetic field mode template. The incident voltage of thej^(th) mode at waveport n can be written as

$\begin{matrix}{{V_{n}^{j}(\omega)} = {\int{\int_{A_{n}}{{e_{n}^{tj}(r)} \times {{\overset{¯}{h}}_{n}^{tj}(r)}{{f(\omega)} \cdot {dA}}}}}} & {{Equation}(81)}\end{matrix}$Divide both sides of Equation (79) by Equation (81) and the square rootof the impedance ratio between waveport m and n, the following results

$\begin{matrix}{{\frac{\int{\int_{A_{m}}{{E_{scat}\left( {r,\omega} \right)} \times {{\overset{¯}{h}}_{m}^{ti}(r)}{{f(\omega)} \cdot {dA}}}}}{\int{\int_{A_{n}}{{e_{n}^{tj}(r)} \times {{\overset{¯}{h}}_{n}^{tj}(r)}{{f(\omega)} \cdot {dA}}}}}.\frac{\sqrt{Z_{n}^{j}(\omega)}}{\sqrt{Z_{m}^{i}(\omega)}}} = {\frac{\int{\int{\int_{V_{obj}}{{{E_{inc}\left( {r^{\prime},\omega} \right)} \cdot {J_{pol}\left( {r^{\prime},\omega} \right)}}{dV}^{\prime}}}}}{\int{\int_{A_{n}}{{e_{n}^{tj}(r)} \times {{\overset{¯}{h}}_{n}^{tj}(r)}{{f(\omega)} \cdot {dA}}}}} \cdot \frac{\sqrt{Z_{n}^{j}(\omega)}}{\sqrt{Z_{m}^{i}(\omega)}}}} & {{Equation}(82)}\end{matrix}$Then the LHS of Equation (82) is the scattered S-parameter betweenwaveport m and n. Substitute Equation (63) the expression ofpolarization current into the RHS of Equation (82), the result is

$\begin{matrix}{{S_{m,n}^{scat}(\omega)} = {\frac{\int{\int{\int_{V_{obj}}{j{\omega\epsilon}_{b}{O\left( r^{\prime} \right)}{{E_{inc}\left( {r^{\prime},\omega} \right)} \cdot {E_{n}\left( {r^{\prime},\omega} \right)}}{dV}^{\prime}}}}}{\int{\int_{A_{n}}{{e_{n}^{tj}(r)} \times {{\overset{¯}{h}}_{n}^{tj}(r)}{{f(\omega)} \cdot {dA}}}}} \cdot \frac{\sqrt{Z_{n}^{j}(\omega)}}{\sqrt{Z_{m}^{i}(\omega)}}}} & {{Equation}(83)}\end{matrix}$In order to acquire the form of Equation (75) and extract the numericalvector Green's function, Equation (83) is reorganized and the final formof the numerical vector Green's function for S-parameter measurementwith waveport excitation is written as

$\begin{matrix}{{G_{m,n}\left( {r^{\prime},\omega} \right)} = \frac{j{E_{inc}\left( {r^{\prime},\omega} \right)}\sqrt{Z_{n}^{j}(\omega)}}{\omega\mu_{0}{\int{\int_{A_{n}}{{e_{n}^{tj}(r)} \times {{\overset{\_}{h}}_{n}^{tj}(r)}{{f(\omega)} \cdot {dA}}\sqrt{Z_{m(\omega)}^{i}}}}}}} & {{Equation}(84)}\end{matrix}$

The numerical vector Green's function above serves to link the objectmaterial contrast function O(r) to the scattered S-parameter between thei^(th) mode of waveport m and j^(th) mode of waveport n. Herein, thequantity E_(inc), Z_(m) ^(i) and Z_(n) ^(j) are obtained numericallywith the 3D FDTD solver, while the quantity e_(n) ^(tj) and h _(n)^(tj)(r) are calculated with the 2D FDFD solver. The key point ofconstructing this numerical vector Green's function is the choice ofJ_(s)(r, t) in Equation (78). The signal waveform f(t) selected toexcite waveport m in Equation (78) should be the same as the one thatexcites waveport n in Equation (80).

E. Validating the Numerical Vector Green's Function

In this section, the steps to validate this numerical vector Green'sfunction with the scattered S-parameter are outlined. First, thescattered S-parameter S_(scat) ^(m,n) with either measurement orsimulation is acquired. Then, the predicted scattered S-parameter usingnumerical vector Green's function with Equation (83) is computed. Thetwo scattered S-parameters to validate the numerical vector Green'sfunction are compared. The detailed step s are listed below.

(1) Obtain the measured scattered S-parameter using Equation (74) witheither measurement or simulation.

(2) Use the 2D FDFD solver to calculate the field mode template forwaveport m and n of the desired modes. The calculated electric andmagnetic field mode template are normalized together with Equation (68)to e^(t)(r) and h^(t)(r), which produce the unit power. Meanwhile, theelectric and magnetic field mode template are normalized individuallywith Equation (72) to ē^(t)(r) and h ^(t)(r), which are used to generatethe waveport voltage and current. Calculate the waveport impedance withthe differential method in Equation (73).

(3) With the presence of the object, we use Equation (69) to excite thej^(th) mode of waveport n with the field mode template e_(n) ^(tj)(r),h_(n) ^(tj)(r) and signal waveform f(t). The total field E_(n)(r′, t) inthe object region is computed and recorded with FDTD.

(4) Without the presence of the object, Equation (77) is used to feed acurrent source J_(s)(r, t) to waveport m. The current source should takethe form of Equation (78) and the signal waveform f(t) used here shouldbe the same as the one in step (3). Then, the incident field E_(inc)(r′,t) within the object region is numerically computed and recorded withFDTD.

(5) With the quantities calculated from step (2) to step (4), Equation(84) is used to construct the numerical vector Green's function and plugit into Equation (75) to predict the scattered S-parameter. Then theestimated scattered S-parameter is compared with the measured scatteredS-parameter from step (1) to validate the numerical vector Green'sfunction.

EXAMPLES AND RESULTS

In this section, the numerical vector Green's function is validated withthree examples. Throughout the examples, the terms measured S-parameteris used to represent the scattered S-parameter acquired with the VNA,simulated S-parameter to mean the scattered S-parameter calculated bythe FDTD simulation, and predicted S-parameter to mean the scatteredS-parameter calculated by the numerical Green's function. The validationsteps were described in detail above.

A. Example 1

First, the numerical vector Green's function is validated in simulationby comparing the simulated S-parameter and the predicted S-parameterfrom two dielectric objects under waveport source excitation. Thesimulation domain is shown in FIG. 41 . In the simulation region, thereare two PEC walls of size 130×70×20 mm³ facing each other 160 mm apart.Within the PEC walls, there are two rectangular apertures. One apertureon the left of FIG. 41 is waveport 1, which has the size of WR-340waveguide and serves as the transmitter. Another aperture on the rightof FIG. 41 is waveport 2, which has the size of WR-430 waveguide andserves as the receiver. Two dielectric objects, which are used togenerate the scattered S-parameter, are placed in the middle of the twowaveports. One is a rectangular box with dimension 40×40 mm² and adielectric constant of 2. On top of the rectangular box, there is adielectric ball with radius of 20 mm and dielectric constant of 4. Theremaining region in the simulation domain is filled with air. The wholesimulation domain is discretized with the hexahedral mesh.

Waveport 1 is excited with its TE10 mode and the TE10 mode scatteredfield in waveport 2 is calculated from 2 GHz to 3 GHz with a frequencystep of 20 MHz. The field mode template for waveport 1 (WR-340) andwaveport 2 (WR-430) are calculated by the 2D FDFD solver at 2.5 GHz. Thesimulated scattered S-parameter is calculated with FDTD using Equation(74) and the S-parameter extraction method described in Navarro et al.The predicted scattered S-parameter is computed with the numericalvector Green's function described herein. First, without the presence ofthe two dielectric objects, the waveport 2 is excited using the TE10mode normalized magnetic field template with the signal waveform, andthe incident field within the object region is computed with FDTD. Then,with the presence of the two dielectric objects, waveport 1 is excitedusing the TE10 mode template of both electric field and magnetic fieldwith the signal waveform, and the total field within the object regionis computed with FDTD. With Equation (83), the predicted scattered S21is obtained.

As can be seen from FIGS. 42A-42B, the predicted scattered S21 matchesvery well with the simulated scattered S21. The amplitude agrees within0.5 dB and the phase agrees within 3 degrees.

B. Example 2

In this example, the numerical vector Green's function in measurementwith a dielectrically loaded two-probe fed rectangular waveguide isvalidated. The waveguide is hollow brass with WR-187 specifications andan outer dimension of 2.54×5.08×15.2 cm³. The two probes are coaxial SMAextended Teflon flange mount connectors. As shown in FIGS. 43A-43B,probe 1 is located 1.91 cm inward from one corner along both axes andprobe 2 is located 2.54 cm and 1.27 cm from the other corner. Here thecoaxial feed of probe 1 is waveport 1, and the coaxial feed of probe 2is waveport 2. Both the excited source at waveport 1 and the receivedfield in waveport 2 are in the TEM mode. The nylon block, which producesthe scattered S-parameter has a dimension of 2.40×1.45×5.05 cm³ and adielectric constant of 3.1.

The measurement is performed with the vector network analyzer from 3 to6 GHz with a frequency step of 50 MHz. The whole waveguide is surroundedby microwave absorber. To measure the scattered S-parameter, the nylonblock is inserted into the bottom left corner of the waveguide asindicated in FIGS. 43A-43B and the S21 is measured with the presence ofthe object. Then, the nylon block is removed and the incident S21 ismeasured. The difference between the two is the scattered S21. Thesimulated scattered S21 is obtained as before with FDTD simulation.

To compute the numerical vector Green's function, the incident and totalelectric field within the object region are computed with the FDTDmethod. The predicted scattered S21 is obtained with the same stepsdescribed above. Note that TEM mode template for both coaxial feed isused, that is the only mode that exists in the measurement frequencyrange. As can b e seen in FIGS. 45A-45B, the amplitude of simulated andpredicted scattered S21 agrees within 0.5 dB, and the phase agreeswithin 2 degrees. The amplitude of the measured and predicted S21 agreeswithin 2 dB, and the phase agrees within 10 degrees.

C. Example 3

In this example, the numerical vector Green's function in measurementwith a square imaging cavity is validated. The imaging cavity is builtfor inverse scattering experiments and has an outer dimension of12.2×12.2×15.2 cm³. The four panels of the square cavity are made ofRoger 3010 substrate with a thickness of 50 mil and dielectric constantof 10.2. There are 36 tapered patch antennas printed on the inside ofthe 4 panels, and their dimensions are shown in FIGS. 46A-46B. Theantennas are designed to radiate into an emulsion fluid couplingmaterial and their resonance frequency is 920 MHz. The measurement isperformed near the resonance frequency of the antenna from 900 to 950MHz with a 2 MHz frequency step. Within this frequency range, theemulsion fluid is measured to have a dielectric constant ϵ_(r) of 19.5and a conductivity σ of 0.075 s/m. A Nylon block with a dimension of5.1×5.1×7.9 cm³ and a dielectric constant of 3.1 is placed in the centerof the imaging cavity as shown in FIGS. 47A-47B. Thus in thisexperiment, the scattered S-parameter is produced by both thepermittivity and conductivity contrast.

In the measurement, the scattered S-parameter was sampled from 3 pairsof antennas, where antenna 1 is used as the transmitter and antennas 2,3, 4 are used as the receivers as shown in FIGS. 47A-47B. Antennas 1, 2,3 are located in the middle ring, while antenna 4 is located in thebottom ring. Four waveports are assigned to the four SMA feeds insimulation. Within the measurement frequency, only TEM mode exists inthe coaxial feed, and thus TEM mode template is used for the waveports.The measured and simulated scattered S-parameters are obtained withEquation (74) and the predicted scattered S-parameter is obtained asdiscussed above.

As can be seen from FIGS. 48A-48B, 49A-49B, and 50A-50B, the amplitudeand phase of the simulated and predicted scattered S-parameter agreeswithin 0.2 dB and 2 degrees. Meanwhile, the amplitude and phase of themeasured and predicted scattered S-parameter agrees within 2 dB and 10degrees.

Various features and advantages of the invention are set forth in thefollowing claims.

What is claimed is:
 1. A method of monitoring in real-time a thermal therapy, the method comprising: positioning an imaging system around a treatment region; imaging the treatment region to obtain dielectric properties; collecting a plurality of pairwise measurements from the treatment region to determine a plurality of S-parameters of the dielectric properties; generating a numerical function directly linking a material property of the treatment region to the plurality of S-parameters, wherein the material property is indicative of a permittivity of the treatment region; determining a solution of the numerical function for the material property of the treatment region using local optimization; mapping the dielectric properties to the treatment region to generate a map of complex permittivity of the treatment region using the solution of the numerical function for the material property; calculating an electric field inside the treatment region and a plurality of calculated S-parameters based on the map of the complex permittivity using a conformal finite-difference time domain (CFDTD) forward solver; generating an updated numerical function based at least in part on the measured electric field: outputting the map of complex permittivity for the treatment region based on the updated numerical function; conducting the thermal therapy on the treatment region based on the map of complex permittivity; generating a thermal image of the treatment region in real-time based on the dielectric properties of the treatment region; and controlling the thermal therapy on the treatment region in response to the thermal image.
 2. The method of claim 1, further comprising acquiring thermal imaging data with a through channel connection to signal a discrete measurement within a continuous data stream and real-time magnitude thresholding to parse the continuous data stream into discrete measurement groups and real-time median filtering within each discrete measurement group.
 3. The method of claim 1, wherein controlling the thermal therapy on the treatment region in response to the thermal image includes a user manually controlling the thermal therapy on the treatment region in response to the thermal image.
 4. The method of claim 1, wherein generating the thermal image of the treatment region relies upon the temperature dependency of the dielectric properties in the treatment region.
 5. The method of claim 1, further comprising positioning the treatment region within a dielectric material prior to conducting the thermal therapy on the treatment region.
 6. The method of claim 1, wherein conducting the thermal therapy on the treatment region includes conducting a microwave thermal therapy on the treatment region. 